Hadamard's Theorem states that: Let $f$ be an entire function of finite order. Denote the zeroes of $f$ by $|a_1|\leq |a_2|\leq \dots$. Then: $f$ gen$f\leq $ord$f\leq $gen$f$+1 where gen is the genus of $f$ and ord is the order of $f$. Furthermore, $f$ can be written in the following way: \begin{equation*} f(z)=e^{Q(z)}z^m \prod_{n=1}^\infty E_{gen (f)}\left(\frac{z}{a_n}\right) \end{equation*} where $Q(z)\in \mathbb{C}[z]$ is a polynomial of degree gen $f$, $m$ is the multiplicity of $f$ at $0$.
So far so good. However, now I read that $\sin(\pi z)=\pi z \prod_{n\in \mathbb Z^*}(1-z/n)e^{z/n}$ and I do not understand the formula:
As ord ($\sin(\pi z))\leq 1$, we know that $Q(z)=a+bz$ for some $a,b\in \mathbb{C}$, however, due to $\sin x/x$ being even and $\sin(\pi z)/z|_{z=0}=\pi$, we get that $e^{Q(z)}=\pi$. So far so good. $\sin(\pi z)$ has a zero of multiplicity 1 at zero, hence the $z$ in the product. Again, so far so good. But now comes the problem and my
Question: We have gen $(\sin(\pi z))=1$. Thus $E_{gen(f)}(z/a_n)=(1-z/n)e^{z/n}$ (due to the zeroes of $\sin(\pi z)$ being the integers. According to the theorem however, $n$ is only taken over the positive integers. In all the formulas, $n$ is taken over all the integers. Why?