Given a hyperbolic 3-manifold with cusps, we consider the composition of a lift of its holonomy in SL(2, C) with the irreducible representation in SL(n, C), that yields a twisted Alexander polynomial An(t), for each natural n. We prove that, for a complex number z with norm one log |An(z)|/n2 converges to the hyperbolic volume of the manifold divided by 4π, as n → ∞. This generalizes and uses a theorem of W. Mueller for closed manifolds on analytic torsion. This is joint work with L. Bénard, J. Dubois and M. Heusener.