Flux through a circle in 3d space

Question

Let $F:\mathbb{R}^3\to\mathbb{R}^3$ be the vector field $F(x,y,z)=0\,\textbf{i}+0\,\textbf{j}+2z\,\textbf{k}$. Now calculate the flux integral $\iint_S F\cdot dS$ upwards (thus facing the positive z direction) where $S$ is the unit circle in the $(x,y)$-axis. Am I wrong to believe that this integral is equal to 0, because $S$ is a flat circle at the origin and thus it has no flux? I'm not able to see how it shouldn't be equal to 0. Is there a way to parameterise $F$ such that this integral is solvable?