Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class

Abstract

Let $\mathcal{L}(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ be an $L$-function in the Selberg class, and $q_{\mathcal{L}}$ its conductor. Let $\ell_0(\mathcal{L})$ be the constant term of the Laurent expansion of $\mathcal{L}'/\mathcal{L}$ at $s=1$. We show that for certain families $\mathcal{F}$ of $L$-functions in the Selberg class with polynomial Euler product: $\bullet$ If $\mathcal{L}\in\mathcal{F}$ has no zeros $\beta + i\gamma$ with $\beta > 1 - \delta(\log q_{\mathcal{L}})^{-1}$, $|\gamma| < (\log q_{\mathcal{L}})^{-1/2}$ for some absolute $\delta >0$, then $\Re(\ell_0(\mathcal{L})) \ll_{\mathcal{F}} \log q_{\mathcal{L}}$; $\bullet$ If $\Re(\ell_0(\mathcal{L})) \ll \log q_{\mathcal{L}}$ for all $\mathcal{L}\in \mathcal{F}$, then there is some absolute $\delta > 0$ such that $\mathcal{L}$ has no zeros $\beta + i\gamma$ with $\beta > 1 - \delta(\log q_{\mathcal{L}})^{-1}$, $|\gamma| < (1-\beta)^{1/2}(\log q_{\mathcal{L}})^{-1/2}$. This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.