Here is the definition of $\xi(s,\chi)$:

$\xi(s,\chi)= \left(\frac{s(s-1)}{2} \right)^{1_{\chi=1}} (q/\pi)^{\frac{s+a}{2}} \Gamma \left( \frac{s+a}{2} \right) L(s,\chi)$

Here is the definition of the **Hadamard product applied to $\xi(s,\chi)$:**

Since it is an entire function of order one, there exists constants $A_\chi, B_\chi \in \mathbb{C}$ such that $\xi(s,\chi)=e^{A_\chi+B_\chi s} \prod_{p} \left(1-\frac{s}{p} \right) e^{s/\rho} $

with the product running over every non-trivial zero of $L(s,\chi)$.

**Why does this imply there are infinitely many non-trivial zeros for $L(s,\chi)?$**

FYI: I know there non trivial zeros are in one to one correspondence with the zeros of $\xi(s,\chi)$