Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

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

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