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## Koecher–Maass series of a certain half-integral weight modular form related to the Duke–Imamoḡlu–Ikeda lift

### Tom 162 / 2014

Acta Arithmetica 162 (2014), 1-42 MSC: Primary 11F67; Secondary 11F46. DOI: 10.4064/aa162-1-1

#### Streszczenie

Let $k$ and $n$ be positive even integers. For a cuspidal Hecke eigenform $h$ in the Kohnen plus space of weight $k-n/2+1/2$ for $\varGamma _0(4),$ let $f$ be the corresponding primitive form of weight $2k-n$ for ${SL}_2(\mathbb {Z} )$ under the Shimura correspondence, and $I_n(h)$ the Duke–Imamoḡlu–Ikeda lift of $h$ to the space of cusp forms of weight $k$ for $Sp_n(\mathbb {Z} )$. Moreover, let $\phi _{I_n(h),1}$ be the first Fourier–Jacobi coefficient of $I_n(h)$, and $\sigma _{n-1}(\phi _{I_n(h),1})$ be the cusp form in the generalized Kohnen plus space of weight $k-1/2$ corresponding to $\phi _{I_n(h),1}$ under the Ibukiyama isomorphism. We give an explicit formula for the Koecher–Maass series $L(s,\sigma _{n-1}(\phi _{I_n(h),1}))$ of $\sigma _{n-1}(\phi _{I_n(h),1})$ expressed in terms of the usual $L$-functions of $h$ and $f$.

#### Autorzy

• Hidenori KatsuradaMuroran Institute of Technology
Mizumoto 27-1
Muroran, 050-8585, Japan
e-mail
Hokkaido University
Kita 10, Nishi 8, Kita-Ku
Sapporo, 060-0810, Japan
and
Department of Mathematics
Hiroshima University
1-7-1 Kagamiyama
Higashi-Hiroshima, 739-8521, Japan
e-mail

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