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)? 



Looking for a special rank 2 vector bundle
Every curve admits a degree g-1 line bundle with $h^0=h^1=0$ -- in fact a generic degree g-1 line bundle has this property, since the space of degree g-1 divisors is g-1 dimensional, but the space of line bundles is g dimensional. Taking the direct sum of two such line bundles will give you a bundle of the type you are seeking.


