Equivalent divisors on Riemann Surfaces

Question

Let $X$ be a compact Riemann surface of genus $g$, it's true that, if $D$ is a effective divisor with degree equal to $g$ then there is $p_1,...,p_g\in X$ such that $D\sim p_1+\cdots+p_g$, i.e there is a meromorphic function $f$ on $X$ s.t $D-(p_1+\cdots+p_g)=div(f)$ where $div(f)$ is the divisor associated to $f$?.