In the context of Dirichlet characters mod $k$, I understand that $L(1, \chi) \ne 0$ for each real-valued non principal character $\chi$ mod $k$, with $L(1, \chi) = \sum_{n=1}^{\infty}{\frac{\chi(n)}{n}}$.
Say I have a general Dirichlet character table for some modulus $k$.
For this table, I'm not sure how to actually verify the result $L(1, \chi) \ne 0$.
I thought that for a Dirichlet character table, all non principal rows must sum to 0? So I'm not sure I see how iterating over each row and summing the real-valued $\phi(k)$ $\chi$s would not also sum to 0?
Apologies if my question is a bit silly, I'm not sure if I fully understand this topic from the book I'm studying from.