Tatuzawa's theorem for Rankin-Selberg $L$-functions

Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $π\otimesχ^2=π$. This extends earlier work of Humphries and Thorner.