Recently, Ono et al. answered problems of Manin by defining zeta-polynomials \(Z_f(s)\) for even weight newforms \(f\in S_k(\varGamma _0(N)\); these polynomials can be defined by applying the “Rodriguez-Villegas transform” to the period polynomial of f. It is known that these zeta-polynomials satisfy a functional equation \(Z_f(s) = \pm \, Z_f(1-s)\) and they have a conjectural arithmetic-geometric interpretation. Here, we give analogous results for a slightly larger class of polynomials which are also defined using the Rodriguez–Villegas transform.

中文翻译：

---

Zeta多项式，Hilbert多项式以及Eichler-Shimura恒等式

最近，小野等。通过为偶数权重新形式\（f \ in S_k（\ varGamma _0（N）\）定义zeta多项式\（Z_f（s）\）来解决Manin的问题；可以通过应用“ Rodriguez-Villegas变换”来定义这些多项式”到的周期多项式˚F。已知的是，这些ζ电多项式满足函数方程\（Z_f（S）= \时\，Z_f（1-S）\） ，并且它们具有推测算术几何解释。在这里，我们为一类较大的多项式给出了类似的结果，这些多项式也使用Rodriguez-Villegas变换定义。