Test vectors for local cuspidal Rankin-Selberg integrals of GL(n), and reduction modulo $\mathrm{\ell }$

Let ${\pi }_1,{\pi }_2$ be a pair of cuspidal complex, or $\mathrm{\ell }$-adic, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\mathrm{\ell }$. Whenever the local Rankin-Selberg $L$-factor $L(X,{\pi }_1,{\pi }_2)$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of ${\pi }_1$ and ${\pi }_2$ such that the local Rankin-Selberg integral associated to these vectors and to the characteristic function of ${\mathfrak{o}}_F^n$ is equal to $L(X,{\pi }_1,{\pi }_2)$. We give an initial application of the test vectors to reduction modulo $\mathrm{\ell }$ of $\mathrm{\ell }$-adic $L$-factors.