  Let $\pi$ and $\pi'$ 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 $L(s,\pi\times\pi')$, 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
$-L'(s,\pi\times\pi')/L(s,\pi\times\pi')$. Also, for $n\leq 8$, we extend the
region of holomorphy and nonvanishing for the twisted symmetric power
$L$-functions $L(s,\pi,\mathrm{Sym}^n\otimes\chi)$ of any cuspidal automorphic
representation of $\mathrm{GL}(2)$.
