Suppose you have a holomorphic line bundle $L$ such that $L^{n}$ is a trivial holomorphic line bundle and the base complex manifold $M$ has no torsion cohomology classes in second degree (i.e. $H^{2}(M, \, \mathbb Z)$ is torsion free).

Then is $L$ holomorphically trivial? If yes, can we remove the restriction on cohomology?