Estimates of automorphic $L$-functions in the discriminant-aspect

Abstract

Let $L(s,\phi)$ be an automorphic $L$-function for a Bianchi group defined via an imaginary quadratic field with discriminant $d < 0$. We give an upper bound for the absolute value of $L(s, \phi)$ in terms of $\Im(s)$, the Laplace eigenvalue and the discriminant $d$. The bound as $d \to \infty$ presents a new aspect in the study of $L$-functions.