Could someone please tell me what results are known about the zeros of L function, $L(s,\chi)$ on the imaginary axis, where $\chi$ is a character mod $q$? In particular, what is known about zeta function? I guess there are none because we know $L(1+it)$ is non zero from the euler product and the functional equation gives $L(it)\neq 0$. Not sure if I am missing something.
Any help would be appreciated. Thanks in advance.
Yes, this turns out to be a fully solved problem. If $\chi$ is a primitive character modulo $q$, then $L(s,\chi)$ has no zeros on the imaginary axis except for a trivial zero at $s=0$ when $\chi$ is an even character. This follows from the functional equation and the fact that $L(1+it,\chi)\ne0$.
If $\chi$ is imprimitive, though, we usually get arithmetic progressions of zeros on the imaginary axis. Suppose that the primitive character $\chi^*$ (mod $q^*$) induces the character $\chi$ (mod $q$). Then we have $$ L(s,\chi) = L(s,\chi^*) \prod_{\substack{p\mid q \\ p\nmid q^*}} \biggl( 1 - \frac{\chi^*(p)}{p^s} \biggr). $$ For each prime $p$ dividing $q$ but not $q^*$, this results in zeros of $L(s,\chi)$ of the form $s = it$ for $t = (\arg\chi^*(p)+2\pi n)/\log p$ for all integers $n$.