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 $F(s,\mathbf{\pi })$ 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 $F(s,\mathbf{\pi })$ with $0<\mathrm{Im}(s)<T$. 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 $F(s,\mathbf{\pi })$ lie near the critical line $\mathrm{Re}(s)=1/2$.

31 pages, 1 figure, Revised version incorporating corrections and clarifications; main conclusions unchanged