The notion of cusp singularities on Hilbert modular surfaces was generalized in higher dimensions by Tsuchihashi by using the theory of toric varieties. In this paper, a dual relation of the invariants of the cusp singularities which was conjectured by \textit{I. Satake} and \textit{S. Ogata} [in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)] is proved. Namely, it is proved that the value at zero \(Z(0)\) of the zeta function associated to an even dimensional cusp singularity is equal to the arithmetic genus defect \(\chi_ \infty\) of its dual cusp singularity. This equality connects the dimension formula of the Hilbert modular cusp forms obtained by the Riemann-Roch theorem and that obtained by the Selberg trace formula.