Let $\chi$ be any primitive Dirichlet character modulo $q \geq 1$. I am looking for a reference for
i) if $L(\rho, \chi) = 0$ and $0 \leq Re(\rho) \leq 1$, then $L(1 - \rho, \chi) = 0$. (and that $\rho$ and $1-\rho$ have the same multiplicities in this case)
ii) $L(s, \chi) \neq 0$ for $Re(s) = 1$.
In Davenport's Multiplicative Number Theory (Page 83) he mentions i) for primitive complex characters, but I haven't been able to find an explicit statement of the above fact in the book.. Any precise reference where I can find these (or an explanation) is appreciated! Thank you!
The functional equation relates $L(s,\chi)$ and $L(1-s,\overline{\chi})$, by $\Lambda(1-s,\chi)=\varepsilon(\chi)\cdot \Lambda(s,\overline{\chi})$, where the $\Lambda$'s are completed $L$-functions, and $|\varepsilon(\chi)|=1$ is the so-called $\varepsilon$-factor (generalizing the argument of a Gauss sum).
Also, using the identity principle from complex analysis, since for real $s>1$ we have $\overline{L(s,\chi)}=L(s,\overline{\chi})$, and $s\to \overline{L(\overline{s},\chi)}$ is holomorphic in $s$, for all $s$ we have $$ \overline{L(\overline{s},\chi)} \;=\; L(s,\overline{\chi}) $$ This gives another symmetry of the zeros, but, again, relating $\chi$ and $\overline{\chi}$.
The only symmetry of zeros mentioning only $\chi$ (and not $\overline{\chi}$ is $s\to 1-\overline{s}$, which is reflection across the line $\Re(s)=1/2$.
For $\chi\not=\overline{\chi}$, we have no reason to think that the zeros of $L(s,\chi)$ are symmetrical under $s\to 1-s$, nor $s\to \overline{s}$, since both those symmetries interchange $\chi$ and $\overline{\chi}$. But applying both does preserve $\chi$, and gives the symmetry across the critical line.