Exceptional zeros of Rankin-Selberg L-functions and joint Sato-Tate distributions

Let be an idele class character over a number field F, and let, ' be non-dihedral twist-inequivalent cuspidal automorphic representations of GL₂ (AF). We prove the following results. 1. If m, n 0 are integers, m+n 1, F is totally real, corresponds with a ray class character, and, ' correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin-Selberg L-function L (s, Symᵐ () (Symⁿ (') ) ) has a standard zero-free region with no exceptional Landau-Siegel zero. When m, n 1 and m+n 4, this is new even for F=Q. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato-Tate distribution for and the joint Sato-Tate distribution for and '. 2. The Rankin-Selberg L-function L (s, Sym² () (Sym² (') ) ) has a standard zero-free region with no exceptional Landau-Siegel zero. Until now, this was only known when =', is self-dual, and is trivial.