Zeros of Polynomials in Derivatives of Automorphic $L$-functions

Abstract

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations $\pi \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbol{\pi})$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the dimensions, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbol{\pi})$ lie near the critical line $\operatorname{Re}(s)=1/2$.