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

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    $\begingroup$ Upper bound: Summation by parts. Lower bound: use the 3-4-1 inequality. $\endgroup$ – Fan Zheng Dec 3 '15 at 18:50
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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, $$ \frac{1}{L(s,\chi)} \ll \log(q(|t|+4)). $$

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