# Global sections of square root line bundle

Let $$C$$ be a smooth curve in $$\mathbb{P}^2$$ over field $$\mathbb{C}$$. Suppose that I have a very ample line bundle $$L$$ on $$C$$ of even degree. Then $$L$$ has $$2^{2g}$$ square roots in $$Pic\ C$$. These are line bundles $$A$$ such that $$A\otimes A=L$$.

What can we say about $$h^0(C,A)$$? Is it non-empty for all $$A$$? Or is it possible that no such $$A$$ has sections?