# Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

## TL;DR

This paper constructs explicit matrix coefficients and Schwartz functions for nonarchimedean GL(n) representations, ensuring the Godement-Jacquet zeta integral equals the associated L-function, thus providing concrete tools for understanding these integrals.

## Contribution

It explicitly identifies choices of data that realize the Godement-Jacquet zeta integral as the L-function for nonarchimedean GL(n) representations.

## Key findings

- Existence of explicit matrix coefficient and Schwartz function choices
- Zeta integral equals the L-function for these choices
- Provides concrete realization of the Godement-Jacquet integral

## 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 $L(s,\pi)$.