The eta function and eta invariant of Z2r-manifolds

We compute the eta function η(s) and its corresponding η-invariant for the Atiyah-Patodi-Singer operator D acting on an orientable compact flat manifold of dimension n=4h−1, h≥1, and holonomy group F≃Z2r, r∈N. We show that η(s) is a simple entire function times L(s,χ4), the L-function associated to the primitive Dirichlet character modulo 4. The η-invariant is 0 or equals ±2k for some k≥0 depending on r and n. Furthermore, we construct an infinite family F of orientable Z2r-manifolds with F⊂SO(n,Z). For the manifolds M∈F we have η(M)=−12|T|, where T is the torsion subgroup of H1(M,Z), and that η(M) determines the whole eta function η(s,M).