Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 4 / c / Solution

There is an absolute $c>0$ such that the product of the Dirichlet L-functions modulo $q$ has no zero inexcept possibly one zero. If it exists, this exceptional zero $\beta$ is real and simple, belongs to a real nonprincipal character ${\chi }_1$, and lies very close to one. Among the primitive characters whose conductors divide $q$, at most one can have such a zero. This is the classical zero-free region for Dirichlet L-functions.