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

Let $\pi_1$ and $\pi_2$ be a pair of cuspidal complex, $\ell$-adic, or banal $\ell$-modular, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\ell$. 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 $L(X,\pi_1,\pi_2)$. As an application, we show that the $\ell$-modular Rankin-Selberg $L$-factor of any two banal cuspidal (hence supercuspidal) $\ell$-modular representations is equal to the reduction modulo-$\ell$ of the $\ell$-adic Rankin-Selberg $L$-factor of any two cuspidal $\ell$-adic lifts. For completeness, we show that the $\ell$-modular Rankin-Selberg $L$-factor of a pair of supercuspidal representations is trivial whenever either one of them is non-banal, thus completing the study of $\ell$-modular supercuspidal Rankin-Selberg $L$-factors.