On special values of standard L-functions of Siegel cusp eigenforms of genus 3

Anh Tuan Do; Kirill Vankov

On special values of standard L-functions of Siegel cusp eigenforms of genus 3

Number Theory 2013-12-24 v1

Authors: Anh Tuan Do , Kirill Vankov

Abstract

We explicitly compute the special values of the standard $L$ -function $L(s, F_{12}, \mathrm{St})$ at the critical points $s\in\{-8, -6, -4, -2, 0, 1, 3, 5, 7, 9\}$ , where $F_{12}$ is the unique (up to a scalar) Siegel cusp form of degree $3$ and weight $12$ , which was constructed by Miyawaki. These values are proportional to the product of the Petersson norms of symmetric square of Ramanujan's $\Delta$ and the cusp form of weight $20$ for ${\rm SL}_2(\mathbb{Z})$ by a rational number and some power of $\pi$ . We use the Rankin-Selberg method and apply the Holomorphic projection to compute these values. To our knowledge this is the first example of a standard $L$ -function of Siegel cusp form of degree $3$ , when the special values can be computed explicitly.

Keywords

Siegel cusp forms standard L-functions special values Miyawaki form Rankin-Selberg method holomorphic projection Petersson norms critical points

Cite

@article{arxiv.1312.6623,
  title  = {On special values of standard L-functions of Siegel cusp eigenforms of genus 3},
  author = {Anh Tuan Do and Kirill Vankov},
  journal= {arXiv preprint arXiv:1312.6623},
  year   = {2013}
}

On special values of standard L-functions of Siegel cusp eigenforms of genus 3

Abstract

Keywords

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