Congruence between Duke-Imamoglu-Ikeda lifts and non-Duke-Imamoglu-Ikeda lifts

Let k and n be positive even integers. For a cuspidal Hecke eigenform g in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(g) be the Duke-Imamoglu-Ikeda lift of g 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 g under the Shimura correspondence. We then characterize prime ideals giving congruence between I(g) and another cuspidal Hecke eigenform not coming from the Duke-Imamoglu-Ikeda lift in terms of the specilal values of the Hecke L-function and the adjoint L-function of f.