We study a class of polynomials that has all of its roots on the critical line and shares many properties with Ehrhart polynomials. Braun showed that the roots of Ehrhart polynomials are bounded quadratically and Higashitani provided examples for polytopes whose Ehrhart polynomial roots come close to this bound. In the case of polytopes which have their roots on the critical line, we present an improved bound. As a side effect this confirms a special case of a conjecture of Braun.