Zeros near $s=1$ and the constant term of $L^{\prime }/L$ 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 ${\mathcal{L}}^{\prime }/\mathcal{L}$ 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.

9 pages, 1 figure

References in corpus (1)

• On Landau-Siegel zeros and heights of singular moduli