Zeros over zeta functions over finite fields

Question

I'm reading about the zeta function over a finite field. The zeta function which is rational function $Z(t)=\frac{P(t)}{(1−t)(1−qt)}$, where $P(t)$ is polynom of degree $2g$ and $g$ is genus. I'm nteresting about their zeros. But reading about this topic I find books and papers that usually writing for of points on a curve over a finite field. Is this the same or what?