Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$

Question: For which $g$, there is such $E$ with $H^0(E)=0$ (and thus $H^1(E)=0$ as well)?