Locally free sheaf generated by global sections and vanishing cohomology on curves

Question

Let $C$ be a smooth projective curve. Let $\mathcal{F}$ be a locally free sheaf on $C$ satisfying $H^1(\mathcal{F})=0$. Is it then true that $\mathcal{F}$ is generated by global sections?

Answer 1

Hint: What about $C=\mathbb{P}^1$, and $\mathcal{F}=\mathcal{O}(-1)$?

Answer 2

If we have $H^1(\mathcal F \otimes \mathcal O(-1))=0$, then $\mathcal F$ is globally generated due to Castelnuovo-Mumford regularity.