text: Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
  @cite_5: We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O K ), where O K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime.
summary: The current paper leaves open a number of challenges. One of them is the computation of the rational functions @math for general @math ; in the special case @math , this has been achieved in @cite_5 . Another one is the computation of the local factors of the @math enumerating subgroups of finite index in @math . This has not even been fully achieved for quadratic number rings @math .