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

Abstract

AbstractWe proof a Garrett–Böcherer decomposition of a vector-valued Siegel Eisenstein series$$E_{l,0}^2$$El,02of genus 2 transforming with the Weil representation of$${\text {Sp}}_2({\mathbb {Z}})$$Sp2(Z)on the group ring$${\mathbb {C}}[(L'/L)^2]$$C[(L′/L)2]. We show that the standard zeta function associated to a vector-valued common eigenformffor the Weil representation can be meromorphically continued to the wholes-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms offand$$E_{l,0}^2$$El,02.