Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

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