Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Sp(n,Z), and f the primitive form of weight 2k-n for SL(2,Z) corresponding to h under the Shimura correspondence. We then express the ratio <I(h),I(h)>/< h, h > of the period of I(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-Imamoglu-Ikeda lift.