TITLE: Is $\frac{1}{L(1+it)}$ unbounded?
QUESTION [9 upvotes]: Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm not sure of general $L$.

REPLY [6 votes]: Yes, this is known to hold, and for more general families of $L$-functions. See in particular Theorem 3.1 and the discussion following it in "Extremal values of Dirichlet $L$-functions in the half-plane of absolute convergence
" by Jörn Steuding.