  We study the eta invariants of Dirac operators and the regularized
determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We
follow Werner M"uller and use relative traces to define these spectral
invariants. We show the regularity of eta and zeta functions at s=0. The
Selberg trace formula and the detailed analysis of the unipotent terms give the
relation between the eta invariant and the Selberg zeta function of odd type.
This allows us to show the vanishing of the unipotent contribution. We show
that the eta invariant and the Selberg zeta function of odd type satisfy
certain functional equation. These results generalize earlier work of John
Millson to hyperbolic manifolds with cusps. We also get the corresponding
relation and functional equation for the regularized determinant and the
Selberg zeta function of even type where the unipotent factor plays a
non-trivial role.
