Line bundles of degree (g-1) on a smooth projective curve

Let $$X$$ be a smooth projective curve of genus $$g>2$$ over an algebraically closed field $$k$$. Does there exist a line bundle $$L$$ on $$X$$ of degree $$(g-1)$$ such that $$H^0(X,L)=0$$?

The locus of $$L$$ with $$H^0(X,L) \ne 0$$ is the image of the natural map $$X^{g-1} \to Pic^{g-1}(X)$$. Its image is at most $$(g-1)$$-dimensional, hence its complement is nonempty.