Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

Abstract

Given an induced representation of Langlands type $(\pi ,V)$ of ${\mathrm{GL}}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $\beta$ and Schwartz-Bruhat function $\Phi$ for which the Godement-Jacquet zeta integral $Z(s,\beta ,\Phi )$ attains the $L$-function $L(s,\pi )$.

7 pages