Abstract
Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$-functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$, in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$.