I am looking for an example of a smooth surface $X$ with $H^1(\mathcal O(k))=0$ for all $k$ (such thing is called an ACM surface, I think) with a globally generated line bundle $L$ such that $L$ is torsion in $Pic(X)$ and $H^1(L) \neq 0$.

Does such surface exist? How can I construct one if it does exist? What if one ask for even nicer surface, such as arithmetically Gorenstein? If not, then I am willing to drop smooth or globally generated, but would like to keep the torsion condition.

To the best of my knowledge this is not a homework question (: But I do not know much geometry, so may be some one can tell me where to find an answer. Thanks.