### Abstract

We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s=12. Specifically, we compute the expected value of L(12,χ) for an ensemble of hyperelliptic curves of genus g over a fixed finite field as g→∞. 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.

Original language | English |
---|---|

Pages (from-to) | 2793-2816 |

Number of pages | 24 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2012 |

### Keywords

- Finite fields
- Function fields
- Hyperelliptic curves
- Moments of quadratic Dirichlet L-functions
- Random Matrix Theory

## Fingerprint Dive into the research topics of 'The mean value of L(12,χ) in the hyperelliptic ensemble'. Together they form a unique fingerprint.

## Cite this

Andrade, J. C., & Keating, J. P. (2012). The mean value of L(12,χ) in the hyperelliptic ensemble.

*Journal of Number Theory*,*132*(12), 2793-2816. https://doi.org/10.1016/j.jnt.2012.05.017