A quick question...
In Section 11.1 of the book of Montgomery & Vaughan's Multiplicative Number Theory when studying the case $χ$ complex it doesn't suppose there can be a real zero for $L(s, χ)$ but when studying the case $χ$ quadratic it says "If $β_0$ is a real zero of $L(s, χ)$, ..."
My question is why $L(s, χ)$ can never be zero for $s$ real and $χ$ complex?
Case 1 of that proof shows that if $\chi$ is complex, then $L(s,\chi)$ does not have a zero in the zero-free region under examination. The proof proceeds by assuming that $\beta_0+i\gamma_0$ is a zero of $L(s,\chi)$ and proving that $\beta \le 1-c/\log q\tau$ (where $\tau=|\gamma_0|+4$). That proof never uses the assumption that $\gamma_0\ne 0$. Therefore Case 1 includes proving that $L(s,\chi)$ does not have a real zero inside the zero-free region when $\chi$ is complex.
