Normal zeta functions of the Heisenberg groups over number rings I: The unramified case

Let K be a number field with ring of integers O K. We compute the local factors of the normal zeta functions of the Heisenberg groups H(O K) at rational primes which are unramified in K. These factors are expressed as sums, indexed by Dyck words, of functions defined in terms of combinatorial objects such as weak orderings. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.