Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$?

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

• Upper bound: Summation by parts. Lower bound: use the 3-4-1 inequality. – Fan Zheng Dec 3 '15 at 18:50

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, $$\frac{1}{L(s,\chi)} \ll \log(q(|t|+4)).$$