quantum field theory - Relation between Noether's charge and the generator of a $U(1)$ symmetry

Consider a $U(1)$ symmetry of a complex scalar field realized as

$$\phi\to\phi^\prime=e^{iQ\theta}\phi.$$ where $Q$ is the generator of the symmetry. The conserved Noether's charge (in $D$-spatial dimensions) corresponding to this symmetry is given by $$Q_N=\int j^0(\textbf{x},t) d^D\textbf{x}=iQ\int d^D\textbf{x}[(\partial_0\phi)\phi^*-(\partial_0\phi^*)\phi].$$ So it turns out that $Q_N\propto Q$ but not equal to $Q$ itself.

But in A. Zee's book QFT in a Nutshell, page 198, the Noether charge corresponding to a symmetry

$$U(\theta)=e^{iQ\theta}$$ is written as $$Q_N=\int j^0(\textbf{x},t)d^D\textbf{x}=Q$$ But as I've shown $Q_N\propto Q$. So is it a mistake there? Or am I missing something?