
531 F.3d 433 (2008)
Mark A. BROWN, Petitioner-Appellant,
v.
Margaret BRADSHAW, Warden, Respondent-Appellee.
No. 06-3482.
United States Court of Appeals, Sixth Circuit.
Argued: April 29, 2008.
Decided and Filed: June 26, 2008.
*435 ARGUED: George C. Pappas, Akron, Ohio, for Appellant. Adam Michael Van Ho, Office of the Ohio Attorney General, Cleveland, Ohio, for Appellee. ON BRIEF: George C. Pappas, Akron, Ohio, Donald J. Malarcik, Gorman, Malarcik & Pierce, Akron, Ohio, for Appellant. Adam Michael Van Ho, Office of the Ohio Attorney General, Cleveland, Ohio, for Appellee.
Before: SILER, BATCHELDER, and SUTTON, Circuit Judges.


*434 OPINION
SILER, Circuit Judge.
Mark A. Brown, an Ohio death row inmate, appeals the decision of the district court denying his petition for a writ of habeas corpus under 28 U.S.C. § 2254. A certificate of appealability ("COA") was granted for one claim: whether clearly established Federal law was violated when the trial court gave an instruction during the penalty phase that allegedly coerced the jurors to agree to recommend a death sentence. We AFFIRM the district court's decision to deny the petition for a writ of habeas corpus because neither the Howard charge nor the post-verdict polling of the jury was a violation of clearly established Federal law.

BACKGROUND
In 1994, Brown murdered Isam Salman and Hayder Al Turk at the Midway Market in Youngstown, Ohio. A jury convicted Brown of two counts of aggravated murder by prior calculation and design. The jury returned guilty verdicts on the firearm specifications and a death penalty specification that the murder was committed as part of a course of conduct involving the purposeful killing of two or more people. During the penalty phase, the jury was instructed to decide which sentence to recommend for each of the two aggravated murders. In the second day of deliberations, the jurors informed the trial court that they had reached agreement on one of the recommendations, but were deadlocked on the other.
The trial court gave the jury the charge approved by the Ohio Supreme Court in State v. Howard, 42 Ohio St.3d 18, 537 N.E.2d 188 (Ohio 1989), which is based on the charge from Allen v. United States, *436 164 U.S. 492, 17 S.Ct. 154, 41 L.Ed. 528 (1896), but modified it slightly to make it appropriate for the penalty phase of a capital trial. The trial court replaced "jurors for acquittal" and "jurors for guilt" with "jurors for life" and "jurors for death." At this point, defense counsel objected that this charge was coercive and suggested an instruction be given similar to the one mentioned in State v. Martens, 90 Ohio App.3d 338, 629 N.E.2d 462 (Ohio Ct.App.1993), informing the jury of the possibility that it might not be able to reach a verdict. The trial court gave only the Howard charge.
The jury later returned with its verdicts: death for the murder of Salman, but life imprisonment for the murder of Al Turk. However, when the jury was polled, juror York indicated that her decision had been a compromise verdict. The trial court reread to the jurors the penalty-phase instructions, minus the Howard charge, and sent the jury back for further deliberations. After two and one-half hours of deliberations, the jury returned with the same verdicts. Upon being polled, all of the jurors said it was their verdict. Accepting the jury's recommendation, the trial court sentenced Brown to death for the murder of Salman, life imprisonment without eligibility for parole for 30 years for the murder of Al Turk, and three years of imprisonment for the firearms specifications.
Brown unsuccessfully sought relief via direct appeal. State v. Brown, 2001 WL 103958 (Ohio Ct.App. Jan. 30, 2001) (unpublished decision), aff'd, 100 Ohio St.3d 51, 796 N.E.2d 506 (Ohio 2003), cert. denied, 540 U.S. 1224, 124 S.Ct. 1516, 158 L.Ed.2d 162 (2004). He also unsuccessfully sought relief via state post-conviction proceedings. State v. Brown, 2003 WL 21518723 (Ohio Ct.App. June 30, 2003) (unpublished decision), juris. denied, 101 Ohio St.3d 1420, 802 N.E.2d 153 (Ohio), cert. denied, 542 U.S. 924, 124 S.Ct. 2880, 159 L.Ed.2d 783 (2004). He submitted an affidavit from juror York stating that after the court's new charge the other jurors yelled and screamed at her in the jury room, pounded the table with their fists, isolated her, accused her of holding things up, blamed her for keeping the other jurors from returning to their families, and bullied her into changing her vote. However, the affidavit was inadmissable. Ohio R. Evid. 606(B).
In 2005, Brown filed a petition for a writ of habeas corpus, raising several claims. The district court denied the petition but granted a COA on the coerced juror claim.

ANALYSIS
We may not grant habeas relief on any claim adjudicated on the merits in State court unless the adjudication resulted in a decision that (1) was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States; or (2) was based on an unreasonable determination of the facts in light of the evidence presented in the State court proceeding. 28 U.S.C. § 2254(d). Clearly established Federal law refers to the holdings, not the dicta, of the Supreme Court at the time of the relevant state court decision. Williams v. Taylor, 529 U.S. 362, 412, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). Brown's primary basis for the clearly established Federal law is Lowenfield v. Phelps, 484 U.S. 231, 241, 108 S.Ct. 546, 98 L.Ed.2d 568 (1988) ("Any criminal defendant, and especially any capital defendant, being tried by a jury is entitled to the uncoerced verdict of that body."). However, Lowenfield does not help Brown.
In Lowenfield, the Supreme Court upheld the denial of a petition for a writ of habeas corpus to a defendant who had been sentenced to death where the combination *437 of giving supplemental instructions and polling the jury was not coercive. Id. at 246, 108 S.Ct. 546. The jury foreman informed the trial court on the second day of deliberations in the sentencing phase that the jury was deadlocked. Id. at 234, 108 S.Ct. 546. The trial court polled the jury on the question of whether "further deliberations would be helpful in obtaining a verdict." Id. Eight jurors anonymously wrote that further deliberations would be helpful while four said they would not. Id. After denying a motion for a mistrial, the trial court received a note that said the jurors had misunderstood the question. Id. The trial court polled the jury again but rephrased his question slightly: "Do you feel that any further deliberations will enable you to arrive at a verdict?" Id. Eleven jurors answered in the affirmative and one answered in the negative. Id. at 234-35, 108 S.Ct. 546. The trial court then instructed the jury that "[e]ach of you must decide the case for yourself" but "[d]o not hesitate to reexamine your own views and to change your opinion if you are convinced you are wrong...." Id. The jury returned 30 minutes later with a verdict of death. Id.
The Supreme Court held that the supplemental charge was not coercive. It approved of the Allen charge, which encouraged the jurors to consider each other's views and "ask themselves whether their own views were reasonable under the circumstances." Id. at 237, 108 S.Ct. 546. Supplemental charges to encourage a verdict are permissible because "[t]he State has in a capital sentencing proceeding a strong interest in having the jury `express the conscience of the community on the ultimate question of life or death.'" Id. at 238, 108 S.Ct. 546 (quoting Witherspoon v. Illinois, 391 U.S. 510, 519, 88 S.Ct. 1770, 20 L.Ed.2d 776 (1968)). Unlike Jenkins v. United States, 380 U.S. 445, 446, 85 S.Ct. 1059, 13 L.Ed.2d 957 (1965), where the trial court told the jury "You have got to reach a decision in this case," the trial court in Lowenfield merely encouraged the jurors to keep deliberating and to consider each other's views.
The Court also held that the polling of the jury was permissible. Unlike Brasfield v. United States, 272 U.S. 448, 449-50, 47 S.Ct. 135, 71 L.Ed. 345 (1926), where the trial court inquired into the numerical division of the jury, the trial court in Lowenfield inquired whether further deliberations might assist the jury in returning a verdict. Lowenfield, 484 U.S. at 240, 108 S.Ct. 546. Asking the jury whether further deliberations would be helpful to reach a verdict without inquiring into the numerical division of the jury is permissible. Id.
Here, the trial court's Howard charge did not violate clearly established Federal law. The Howard charge is no more coercive than the Allen charge. As in Allen and Lowenfield, the supplemental charge given here merely encouraged the jurors to consider each other's views and to ask themselves whether their own views were reasonable under the circumstances. See Lowenfield, 484 U.S. at 237-38, 108 S.Ct. 546. Such a supplemental instruction is permissible because it is not coercive and because it respects the state's strong interest in "having the jury express the conscience of the community on the ultimate question of life or death." Id. at 238, 108 S.Ct. 546 (internal quotations omitted). The trial court, in short, did not order the jury to reach a verdict.
The polling of the jury under these circumstances also did not violate clearly established Federal law. Lyell v. Renico, 470 F.3d 1177, 1183 (6th Cir.2006) (finding no violation of clearly established Federal law when trial court conducted post-verdict juror polling). While the trial *438 court exposed the numerical division of the jurors, "there is a world of difference between juror-coercion claims arising from deadlocked juries and those arising from post-verdict juror polling," thus distinguishing Brasfield. Id. Coercion does not happen whenever the trial court, consistent with a request for polling by the defendant, requires the juror to make a public statement of the juror's position after the verdict has been announced. Id. "The prospect that juror 12, who retained the courage of her convictions to tell the public that she now wanted to change her vote, would be browbeaten into submitting to the majority ... seems quite slim." Id. at 1184. The post-verdict polling did not violate clearly established Federal law.
Brown argues that when the jury could not reach a verdict, the trial court should have taken the case from the jury and imposed a life sentence under the procedure discussed in State v. Springer, 63 Ohio St.3d 167, 586 N.E.2d 96, 100 (Ohio 1992), rather than giving the Howard charge. This issue is governed by state law, and we may not issue the writ on the basis of a perceived error of state law. Pulley v. Harris, 465 U.S. 37, 41, 104 S.Ct. 871, 79 L.Ed.2d 29 (1984).
Brown argues that the alleged coercion of juror York violated Mills v. Maryland, 486 U.S. 367, 384, 108 S.Ct. 1860, 100 L.Ed.2d 384 (1988), by precluding the consideration of mitigating factors. Brown did not raise this argument in the state courts. Therefore, it is procedurally defaulted. Maupin v. Smith, 785 F.2d 135, 138-39 (6th Cir.1986).
We deny Brown's request for discovery and an evidentiary hearing on the issue of juror coercion. Discovery and an evidentiary hearing on this issue would be futile because such evidence is inadmissible under Ohio law, Ohio R. Evid. 606(B), and Brown has not established any constitutional impediment to enforcing this state evidentiary rule.
AFFIRMED.
