Let $\chi_1$ be the real character associated with the exceptional zero $\beta$. The <prime number theorem in an arithmetic progression with an exceptional zero> gives, uniformly for $x\geq2$ and $(a,q)=1$,
$$
\boxed{
\psi(x;q,a)
=\frac{x}{\varphi(q)}
-\frac{\chi_1(a)x^\beta}{\beta\varphi(q)}
+O\left(
x(\log q)^2
\exp\left[-\frac{c\log x}{\log q+\sqrt{\log x}}\right]
\right).
}
$$
If no exceptional zero exists, the middle term is omitted. The constant $c>0$ is absolute.

Solution

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

Uniformly for $x\geq2$ and $(a,q)=1$,
$$
\psi(x;q,a)
=\frac{x}{\varphi(q)}
-\frac{\chi_1(a)x^\beta}{\beta\varphi(q)}
+O\left(
x(\log q)^2
\exp\left[-\frac{c\log x}{\log q+\sqrt{\log x}}\right]
\right),
$$
where the second term occurs only when a real character $\chi_1$ has an <exceptional zero> $\beta$.

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 Incoming links: Prime number theorem in an arithmetic progression with an exceptional zero