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

Abstract

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\frac{1}{2}$. Specifically, we compute the expected value of $L(\frac{1}{2},\chi )$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\to \infty$. 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.

arXiv comment22 pages, To appear in Journal of Number Theory Volume 132, Issue 12, December 2012, Pages 2793-2816

References in corpus (3)

• Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
• Nonvanishing of quadratic Dirichlet L-functions at s=1/2
• Statistics for traces of cyclic trigonal curves over finite fields

Cited by in corpus (15)

• Conjectures for the integral moments and ratios of L-functions over function fields
• Moments of quadratic twists of elliptic curve L-functions over function fields
• Complex Moments and the distribution of Values of $L(1,{\chi }_D)$ over Function Fields with Applications to Class Numbers
• Moments of zeta functions associated to hyperelliptic curves over finite fields
• Hybrid Euler-Hadamard product for quadratic Dirichlet $L$-functions in function fields
• Newman's conjecture in various settings
• Moments of quadratic Dirichlet $L$-functions over Function Fields
• A representation theory approach to integral moments of L-functions over function fields
• Rudnick and Soundararajan's Theorem for Function Fields
• Hybrid Euler-Hadamard Product for Dirichlet $L$-functions with Prime conductors over Function Fields
• Average values of L-functions in even characteristic
• Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field
• Average Values of $L$-series for Real Characters in Function Fields
• A Simple Proof of the Mean Value of $\left|K_2(\mathcal{O})\right|$ in Function Fields
• The First Moment of $L(\frac{1}{2},\chi )$ for Real Quadratic Function Fields