An invariant for non simply connected manifolds

For a closed manifold $M$ we introduce the set of co-Euler structures and we define the modified Ray-Singer torsion, a positive real number associated to $M,$ a co-Euler structure and an acyclic representation $\rho$ of the fundamental group of $M$ with $H^\ast(M;\rho)=0.$ If the co-Euler structure is integral we show that the modified Ray--Singer torsion, regarded as a positive (real valued) function on the variety of some complex representations, is the absolute value of a (complex valued) rational function which carries interesting topological information about the manifold. This rational function is the invariant in the title. If the co-Euler structure is arbitrary one obtains a more general object, a holomorphic 1-cocycle. Interesting rational functions in topology appear in this way.The argument of this rational function when defined, is an interesting and apparently unexplored invariant which reminds the Atiyah--Patodi--Singer eta invariant.