A Selberg-type zero-density result for twisted $\mathrm{G}{\mathrm{L}}_2$ $L$-functions and its application

Abstract

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $\ddot{I}{‡}_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $\ddot{I}‡$ be a primitive Dirichlet character modulus $q$ with $\ddot{I}‡\ne \ddot{I}{‡}_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s,f\otimes \ddot{I}‡)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t,f\otimes \ddot{I}‡)=\ddot{I}{€}^{-1}\arg L(1/2+\ddot{A}\pm t,f\otimes \ddot{I}‡)$ and prove a central limit theorem for its distribution over $\ddot{I}‡$ of modulus $q$.

23 pages