Assume that you are given a smooth projective algebraic variety $X$ and a divisor $D$ with normal crossings on it. Put $U=X-D$. Assume now that $L$ is a torsion line bundle(=invertible sheaf) on $U$, i.e. there exists an integer $N$ such that $L^{\otimes N}=\mathcal{O}_U$. I have the following question:

*How to extend $L$ to a line bundle $L_X$ on $X$ such that $L_X^N=\mathcal{O}_X(D)$?*

Will the direct image with compact support $j_! L$ with respect to the inclusion $j: U \hookrightarrow X$ be such an extension?

I have the feeling that this should be pretty obvious what I cannot see how.

Thanks for your help