Let $n,m$ be two different positive integers.
I heard that for cuspidal tempered automorphic representations $\pi_{n}$ and $\pi_m$ of $GL_n$ and $GL_m$, the Rankin-Selberg L-function $L(s,\pi_n \times \pi_m)$ is entire on $\mathbb{C}$.
I am wondering if the tempered or cuspidal condition is essential to ensure $L(s,\pi_n \times \pi_m)$ being entire. That means if one of $\pi_n$ or $\pi_m$ is non-tempered or non-cuspidal, then their $L$-function may have a pole? If this is the case, can we know the candidate for the location of poles?