A zero density estimate and fractional imaginary parts of zeros for <i>L</i>-functions

Abstract We prove an analogue of Selberg’s zero density estimate for $\zeta(s)$ that holds any $\textrm{GL}_2$ L -function. use this to study the distribution vector fractional parts $\gamma\boldsymbol{\alpha}$ , where $\boldsymbol{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over imaginary nontrivial zeros a