  This is the first of two papers in which we introduce and study two bivariate
zeta functions associated to unipotent group schemes over rings of integers of
number fields. One of these zeta functions encodes the numbers of isomorphism
classes of irreducible complex representations of finite dimensions of
congruence quotients of the associated group and the other one encodes the
numbers of conjugacy classes of each size of such quotients. In this paper, we
show that these zeta functions satisfy Euler factorizations and almost all of
their Euler factors are rational and satisfy functional equations. Moreover, we
show that such bivariate zeta functions specialize to (univariate) class number
zeta functions. In case of nilpotency class 2, bivariate representation zeta
functions also specialize to (univariate) twist representation zeta functions.
