How we can show $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$?

Question

We know that the functional equation for Riemann-Zeta is $\psi(s)=\frac{1}{2}s(s-1)\pi^{-1/2s}\Gamma(1/2s) \zeta(s)$ and $\psi(s) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$. Does it make sense to say $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$ since $\zeta(s)$ is the trivial case for $L(s,\chi)$ and $L(s,\chi)$ attains its maximum value when $\chi= 1$ or, when $L(s,\chi) = \zeta(s)?$ (Here $\Lambda(s,\chi)$ is the completed $L$-series.) I want to use the fact $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ in order to apply Phragmen-Lindelöf Principle on $L(s,\chi)$. I appreciate any help!

Answer 1

For $s\not \in (1,\infty)$ "attains its maximum value for $\chi=1$" doesn't make sense; The proof of $\le \exp(C |s|\log |s|)$ is the same for Dirichlet L-functions, showing that they are $O(|s|)$ on $\Re(s)\ge 1/2$ and then extending to the whole complex plane with the functional equation and Stirling's formula for $\Gamma(s)$.