Koecher–Maass series of a certain half-integral weight modular form related to the Duke–Imamoḡlu–Ikeda lift
Volume 162 / 2014
Abstract
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 $f$.
