About the space of divisor $L(D):=\{f: \text{meromorphic}, (f)+D\geq 0\}$

Question

Q1: I have known that: If $M$ be a compact Riemann surface. If $D$ is a divisor on $M$ with $d(D)<0$, then $L(D)=\{0\}$.

But how about the situation if remove the condition "compact"?

Q2: what is the dimension $\dim_{\mathbb{C}} L(D)$ is equal under the complex field $\mathbb{C}$?