  Let M be an oriented irreducible 3-manifold with infinite fundamental group
and empty or toroidal boundary. Consider any element \phi in the first
cohomology of M with integral coefficients. Then one can define the
\phi-twisted L^2-torsion function of the universal covering which is a function
from the set of positive real numbers to the set of real numbers. By earlier
work of the second author and Schick the evaluation at t=1 determines the
volume.
  In this paper we show that its degree, which is a number extracted from its
asymptotic behavior at 0 and at infinity, agrees with the Thurston norm of
\phi.
