In Davenport, Multiplicative Number Theory, page 6, the author asserts that:
"Let $\omega$ be a complex character. Suppose $L_{\omega}(1)=0$, then it would imply that $L_{\omega}(s)=L_{\omega}(s)-L_{\omega}(1) = (s-1)L'_{\omega}(s_1)$, where $s>1$ and $1<s_1<s$."
I do not think that we are allowed to use MVT here, as $L_{\omega}(s)$ does not have to be a real-valued function for every $s>1$. (or this is the piece I could not figure out.)
I looked at the Euler product expression for $\displaystyle L_{\omega}(s) = \prod_{p\neq q}\left(1-\frac{\omega^{v(p)}}{p^s}\right)^{-1}$, but it does not really seem to be real for every $s>1$.
I tried a complex character $\mathrm{mod}7$ with the character $\omega$ respectively taking the values $1,e^{2\pi i/3},e^{\pi i/3},e^{-\pi i/3},e^{-2\pi i/3},-1$ and I obtained with the help of wolfram alpha that $L_{\omega}(2) \approx 0.902 - 0.242i$. Since Dirichlet $L$ functions are absolutely convergent for $s>1$, I separated the modulo classes and add them up later on. Is my procedure faulty?
I also know that $\displaystyle\prod_{\omega} L_{\omega}(s) \geq 1$ for $s>1$, but from this I cannot deduce anything. Is there some kind of another tool the author is using here or is the $L$ function real for $s>1$ for some mysterious reason I am not aware of? Thanks in advance.