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?