On analytic properties of the standard zeta function attached to a vector valued modular form

Abstract

We proof a Garrett-Böcherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of ${\text{Sp}}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L^{\prime }/L{)}^2]$. We show that the standard zeta function associated to a vector valued common eigenform $f$ for the Weil representation can be meromorphically continued to the whole $s$-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of $f$ and $E_{l,0}^2$.

28 pages, submitted for publication