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arXiv · 2305.15120

Expected Values of $L$-functions Away from the Central Point

Chantal David·Patrick Meisner
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Abstract

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindelöf hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.

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Chantal David, Patrick Meisner. 2025-02-06. Expected Values of $L$-functions Away from the Central Point. https://arxiv.org/abs/2305.15120

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