A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application

Abstract

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