$-\operatorname{Re}\,L'/L(s,\chi)\le c\log(q(|t|+2))-\sum_{\rho}\operatorname{Re}\frac{1}{s-\rho}$ near $\operatorname{Re}s=1$ (Davenport §14)

The zero-sum bound for $-L'/L$ near the line $\operatorname{Re}s=1$ (Davenport §14, from the Hadamard product of §12). There is an absolute constant $c>0$ such that for every modulus $q\ge1$, every non-principal Dirichlet character $\chi$ modulo $q$, every $s=\sigma+it$ with $1<\sigma\le2$, and every finite multiset $Z$ of zeros of $L(\cdot,\chi)$ lying in the closed disc $|\rho-s|\le\tfrac12$, each zero repeated at most as often as its multiplicity,

Davenport proves, for primitive $\chi$, the identity $-L'/L(s,\chi)=\tfrac12\log\tfrac q\pi+\tfrac12\tfrac{\Gamma'}{\Gamma}\bigl(\tfrac{s+a}{2}\bigr)-B(\chi)-\sum_\rho\bigl(\tfrac1{s-\rho}+\tfrac1\rho\bigr)$ with $\operatorname{Re}B(\chi)=-\sum_\rho\operatorname{Re}\tfrac1\rho$, giving $-\operatorname{Re}L'/L(s,\chi)<c\log(q(|t|+2))-\sum_\rho\operatorname{Re}\tfrac1{s-\rho}$ over all nontrivial zeros; since every term $\operatorname{Re}\tfrac1{s-\rho}$ is positive for $\operatorname{Re}s>1$, the sum may be restricted to any sub-multiset of the zeros, in particular to those within distance $\tfrac12$ of $s$ (a local form that can also be obtained by the Borel–Carathéodory method without the Hadamard product). Imprimitive $\chi$ reduce to the inducing primitive character, whose extra Euler factors contribute $O(\log q)$ and no zeros with $\operatorname{Re}\rho>0$. Multiplicities are expressed through Mathlib's analyticOrderAt. This is the central analytic input of both the zero-free region and the estimates for $\psi(x,\chi)$.