Value-Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic $L$-functions

Abstract

Let $d\in\mathbb{N}$, and let $\pi$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,\pi\otimes\chi_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $\chi_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}(1,\pi\otimes\chi_D)\right|$ when $\pi$ is self-dual.