  We investigate how one can twist L^2-invariants such as L^2-Betti numbers and
L^2-torsion with finite-dimensional representations. As a special case we
assign to the universal covering of a finite connected CW-complex X together
with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0}
to R, provided that the fundamental group of X is residually finite and its
universal covering is L^2-acyclic.
