I would guess this is some standard fact related to the zero-free region. But cannot find it in the textbooks I read.



Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$?
For starters, see Montgomery & Vaughan's "Multiplicative Number Theory", Theorem 11.3 and 11.4. For example, in the zero free region of 11.3,
$$
\frac{1}{L(s,\chi)} \ll \log(q(|t|+4)).
$$


