Let G be a finite group , p a prime number and n a positive integer. Following L. Solomon, one can define a zeta function of the free Z_p[G]-module of rank n (where Z_p is the ring of p-adic integers), counting submodules of finite index. We present a functional equation for this zeta function, extending the proof of C.J. Bushnell and I. Reiner for n=1, in order to cover our more general situation. In addition, some examples and explicit formulas are considered.