Functional equations for local normal zeta functions of nilpotent groups

We give an explicit formula for the local normal zeta functions of torsion-free, class-2-nilpotent groups at primes p where the associated Pfaffian hypersurface has good reduction mod p and contains no lines. We show how together the functional equations of two types of zeta functions the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa may give rise to functional equations for local normal zeta functions of groups, confirming a conjecture of du Sautoy. We also give explicit formulae and derive functional equations for a family of class-2-nilpotent groups known as Grenham groups.