LAUSR

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$ be a pair of cuspidal complex, or $\ell$ -adic, representations of the general linear group of rank $n$ over a nonarchimedean local field $F$ of residual characteristic $p$ , different to $\ell$ . Whenever the local Rankin–Selberg $L$ -factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$ is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ . As an application we prove that the $L$ -factor of a pair of banal $\ell$ -modular cuspidal representations is the reduction modulo $\ell$ of the $L$ -factor of any pair of $\ell$ -adic lifts.