Does the Weil conjecture work with 0 dimensional varieties?

Question

Suppose I have some inseparable quadratic polynomial $p(x)$ over $\mathbb{F}_q$ which has a pair of roots in $\mathbb{F}_{q^2}$. If I compute the zeta function of the variety cut out by $p(x)$ (in either $\mathbb{A}^1$ or $\mathbb{P}^1$, sinc the number of points will be the same) I get $$\zeta(X,t)=\exp\left(\sum_{n=2}^{\infty}\frac{2}{n}t^n\right)=\frac{e^{-2t}}{1-t}$$ which is not a rational function. What does this mean then for quasi-projective varieties with non-rational points deleted (like $\mathbb{P}^1$ minus the roots of $p(x)$)? I feel like I'm missing something obvious.

Answer 1

That's not the zeta function. There are two points over $\mathbb{F}_{q^n}$ if $n$ is even and none if $n$ is odd. This means the zeta function is

$$\exp \left( \sum_{k \ge 1} \frac{2}{2k} t^{2k} \right) = \frac{1}{1 - t^2}.$$