Let k and n be positive even integers. For a cus- pidal Hecke eigenform h in the Kohnen plus subspace of weight k − n/2 + 1/2 for 0(4), let In(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Spn(Z), and f the primitive form of weight 2k − n for SL2(Z) corresponding to h under the Shimura correspondence. We then express the ratio hIn(h),In(h)i/hh,hi 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- Imamoglu-Ikeda lift. One of the fascinating problems in the theory of modular forms is to find the relation between the periods (or the Petersson products) of cuspidal Hecke eigenforms which are related with each other through their L-functions. In particular, there are several important results concerning the relation between the period of a cuspidal Hecke eigen- form g for an elliptic modular group Γ � SL2(Z) and that of its lift bHere, by a lift of g we mean a cuspidal Hecke eigenform for another modular group Γ 0 (e.g. the symplectic group, the orthogonal group, the unitary group, etc.) whose certain L-function can be expressed in terms of certain L-functions related with g:Thus we propose the following problem: Problem A. Express the ratio hbbi=hg;gi e in terms of arithmetic invariants of g;for example, the special values of certain L-functions related with g for some integer e: For instance, Zagier (Zag77) solved Problem A for the Doi-Naganuma