Conditions on $\mathcal{F}$ such that $\chi(\mathcal{F}) = 0$ for a coherent sheaf on a curve over $k$.

Question

Let $X$ be a projective curve (you may also assume Cohen-Macaulayness) over some field $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then the Euler-Poincare characteristic of $\mathcal{F}$ is equal to $$ \chi(\mathcal{F}) = \dim_k H^0(X,\mathcal{F}) - \dim_k H^1(X,\mathcal{F}). $$

What are necessary and what are sufficient conditions on $\mathcal{F}$ such that $\chi(\mathcal{F}) = 0$?