On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points

Let $Q(z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j(z)$ traverse an orthonormal basis of Hecke–Maass cusp forms of full level. Let $1\mathrm{/}4+t_j^2$ be the Laplace eigenvalue $u_j(z)$. In this paper, we prove that there is a constant $\gamma (Q)$ expressed as a certain Euler product associated to $Q$ such that at least $\gamma (Q)\mathrm{/}11$ of the Rankin–Selberg special $L$-values $L(1\mathrm{/}2+i{t}_j,Q\otimes u_j)$ for $t_j⩽T$ do not vanish as $T\to \mathrm{\infty }$. Further, we show that the non-vanishing proportion is at least $\gamma (Q)\cdot (4\mu -3)\mathrm{/}(4\mu +7)$ on the short interval $\mathrm{\mid }{t}_j-T\mathrm{\mid }⩽T^{\mu }$ for any $3\mathrm{/}4<\mu <1$.