# Test vectors for Rankin-Selberg $L$-functions

## TL;DR

This paper constructs explicit test vectors for local Rankin-Selberg $L$-functions associated with pairs of generic representations of $GL_n imes GL_m$ over $p$-adic fields, facilitating the computation of local zeta integrals.

## Contribution

It introduces a unipotent averaging method to produce Whittaker functions that serve as test vectors for local $L$-functions, expressed explicitly in terms of Langlands parameters.

## Key findings

- Constructed non-zero local zeta integrals using unipotent averaging.
- Expressed zeta integrals explicitly via Langlands parameters.
- Provided conditions under which Whittaker functions serve as test vectors.

## Abstract

We study the local zeta integrals attached to a pair of generic representations $(\pi,\tau)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $\pi$ and $\tau$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.