  To a closed Riemannian manifold, we associate a set of (special values of) a
family of Dirichlet series, indexed by functions on the manifold. We study the
meaning of equality of two such families of spectral Dirichlet series under
pullback along a map. This allows us to give a spectral characterization of
when a smooth diffeomorphism between Riemannian manifolds is an isometry, in
terms of equality along pullback. We also use the invariant to define the
(spectral) length of a map between Riemannian manifolds, where a map of length
zero between manifolds is an isometry. We show that this length induces a
distance between Riemannian manifolds up to isometry.
