We consider a certain Dirichlet series of Rankin-Selberg type associated with two Siegel cusp forms of the same integral weight with respect to Sp n (Z). In particular, we give an explicit formula for the Dirichlet series associated with the Ikeda lifting of cuspidal Hecke eigenforms with respect to SL 2 (Z). We also comment on a contribution to the Ikeda's conjecture on the period of the lifting.

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 Γ0(4), let In(h) be the Duke-Imamoḡlu-Ikeda lift of h in the space of cusp forms of weight k for Sp n (Z), and f be the primitive form of weight 2k − n for SL2(Z) corresponding to h under the Shimura correspondence. We then express the ratio In(h), In(h) / h, h of the period of In(h) to that of h in terms of special values of certain L-functions of f . This proves the conjecture proposed by Ikeda concerning the period of the Duke-Imamoḡlu-Ikeda lift.be the character corresponding to Q( √ D)/Q. Here we make the convention that D * = 1 if D = 1.

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 Γ 0 (4), let f be the corresponding primitive form of weight 2k − n for SL 2 (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 (Z). Moreover, let φ In(h),1 be the first Fourier-Jacobi coefficient of I n (h) and σ n−1 (φ In(h),1 ) be the cusp form in the generalized Kohnen plus space of weight k − 1/2 corresponding to φ In(h),1 under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series L(s, σ n−1 (φ In(h),1 )) of σ n−1 (φ In(h),1 ) expressed in terms of the usual L-functions of h and f .

No abstract

No abstract