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arXiv · 1208.1131

The Mean Value of $L(\tfrac{1}{2},χ)$ in the Hyperelliptic Ensemble

J. C. Andrade·J. P. Keating
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Abstract

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\tfrac{1}{2},χ)$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\rightarrow\infty$. Our approach relies on the use of the analogue of the approximate functional equation for such $L$--functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of $L$--functions motivated by Random Matrix Theory.

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J. C. Andrade, J. P. Keating. 2012-08-06. The Mean Value of $L(\tfrac{1}{2},χ)$ in the Hyperelliptic Ensemble. https://doi.org/10.1016/j.jnt.2012.05.017

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