Summary: Let \(k\) and \(n\) be positive even integers. For a cuspidal Hecke eigenform \(g\) in the Kohnen plus subspace of weight \(k-n/2+1/2\) for \(\Gamma_0(4)\), let \(f\) be the primitive form of weight \(2k-n\) for SL\(_2(\mathbb Z)\) corresponding to \(g\) under the Shimura correspondence, and let \(I_n(g)\) be the Duke-Imamoglu-Ikeda lift of g in the space of cusp forms of weight k for Sp\(_n(\mathbb Z)\). Then we characterize prime ideals giving congruence between \(I_n(g)\) and another cuspidal Hecke eigenform not coming from the Duke-Imamoglu-Ikeda lift in terms of the special values of the Hecke \(L\)-function and the adjoint \(L\)-function of \(f\).