  In this paper, we study lower bounds of a general family of $L$-functions on
the $1$-line. More precisely, we show that for any $F(s)$ in this family, there
exists arbitrary large $t$ such that $F(1+it)\geq e^{\gamma_F} (\log_2 t +
\log_3 t)^m + O(1)$, where $m$ is the order of the pole of $F(s)$ at $s=1$.
This is a generalization of the same result of Aistleitner, Munsch and the
second author for the Riemann zeta-function. As a consequence, we get lower
bounds for large values of Dedekind zeta-functions and Rankin-Selberg
$L$-functions of the type $L(s,f\times f)$ on the $1$-line.
