characters on $\mathbb Z_l$ into $\mathbb C_p^*$

Question

Suppose I have a character $$ \chi : \prod _{l \not | p} \mathbb Z_l^* \rightarrow \mathbb C_p ^*$$ where $\mathbb C_p$ is the p adic completion of algebraic closure $\overline {\mathbb Q _p}$ of $\mathbb Q_p$.
How does one prove that this character factors through a finite quotient?