  We give explicit formulae for the local normal zeta functions of
torsion-free, class-2-nilpotent groups, subject to conditions on the associated
Pfaffian hypersurface which are generically satisfied by groups with small
centre and sufficiently large abelianization. We show how 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
- match up to give a functional equation for local normal zeta functions of
groups. We also give explicit formulae and derive functional equations for an
infinite family of class-2-nilpotent groups known as Grenham groups, confirming
conjectures of du Sautoy.
