  We define the rank-metric zeta function of a code as a generating function of
its normalized $q$-binomial moments. We show that, as in the Hamming case, the
zeta function gives a generating function for the weight enumerators of
rank-metric codes. We further prove a functional equation and derive an upper
bound for the minimum distance in terms of the reciprocal roots of the zeta
function. Finally, we show invariance under suitable puncturing and shortening
operators and study the distribution of zeroes of the zeta function for a
family of codes.
