I am self studying number theory from Tom Apostol's book and I have a question in proof of theorem 12.9 on page 262.
If $\psi$ is not principal then $ L(s, \chi) $ will be analytic for all s as product p|k is finite and L(s, $\psi$) will be analytic for all s as $\psi$ is not principal character.
But I have no reason to believe that $\psi$ is not principal. Can you tell if there is some reasoning which I am missing or if my way of proving is wrong, kindly tell the right way.
Thanks!!