A new zero-free region for Rankin-Selberg $L$-functions

Let $\pi$ and ${\pi }^{\mathrm{\prime }}$ be cuspidal automorphic representations of $\mathrm{G}\mathrm{L}(n)$ and $\mathrm{G}\mathrm{L}(n^{\mathrm{\prime }})$ with unitary central characters. We establish a new zero-free region for all $\mathrm{G}\mathrm{L}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\mathrm{\prime }})$, generalizing Siegel’s celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L^{\mathrm{\prime }}(s,\pi \times {\pi }^{\mathrm{\prime }})\mathrm{/}L(s,\pi \times {\pi }^{\mathrm{\prime }})$. Also, for $n\le 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,\pi ,{\mathrm{S}\mathrm{y}\mathrm{m}}^n\otimes \chi )$ of any cuspidal automorphic representation of $\mathrm{G}\mathrm{L}(2)$.