  We study zeta functions enumerating submodules invariant under a given
endomorphism of a finitely generated module over the ring of ($S$-)integers of
a number field. In particular, we compute explicit formulae involving Dedekind
zeta functions and establish meromorphic continuation of these zeta functions
to the complex plane. As an application, we show that ideal zeta functions
associated with nilpotent Lie algebras of maximal class have abscissa of
convergence $2$.
