### Abstract

We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a critical special value of L(s, F \times g). This is in the spirit of known results on critical values of triple product L-functions, also of degree 8, though there are significant differences.

Original language | English |
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Pages (from-to) | 1773-1837 |

Number of pages | 65 |

Journal | International Mathematics Research Notices |

Volume | 2009 |

Issue number | 10 |

Early online date | 13 Feb 2009 |

DOIs | |

Publication status | Published - 2009 |

### Bibliographical note

48 pages, typos corrected, some changes in Sections 6 and 7, other minor changes### Keywords

- math.NT
- math.RT
- 11F46; 11F67; 11F70; 22E50; 22E55;

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## Cite this

Saha, A. (2009). L-functions for holomorphic forms on GSp(4) x GL(2) and their special values.

*International Mathematics Research Notices*,*2009*(10), 1773-1837. https://doi.org/10.1093/imrn/rnp001