Let G G be a finitely generated nilpotent group. The representation zeta function ζ G ( s ) \zeta _G(s) of G G enumerates twist isoclasses of finite-dimensional irreducible complex representations of G G . We prove that ζ G ( s ) \zeta _G(s) has rational abscissa of convergence α ( G ) \alpha (G) and may be meromorphically continued to the left of α ( G ) \alpha (G) and that, on the line { s ∈ C ∣ R e ( s ) = α ( G ) } \{s\in \mathbb {C} \mid \mathrm {Re}(s) = \alpha (G)\} , the continued function is holomorphic except for a pole at s = α ( G ) s=\alpha (G) . A Tauberian theorem yields a precise asymptotic result on the representation growth of G G in terms of the position and order of this pole. We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form G ( O ) \mathbf {G}(\mathcal {O}) , where G \mathbf {G} is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring O \mathcal {O} of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of G \mathbf {G} , independent of O \mathcal {O} .