  In this paper, we investigate the zeros near the critical line of linear
combinations of $L$-functions belonging to a large class, which conjecturally
contains all $L$-functions arising from automorphic representations on
$\text{GL}(n)$. More precisely, if $L_1, \dots, L_J$ are distinct primitive
$L$-functions with $J\ge 2$, and $b_j$ are any nonzero real numbers, we prove
that the number of zeros of $F(s)=\sum_{j\leq J} b_j L_j(s)$ in the region
$\text{Re}(s)\geq 1/2+1/G(T)$ and $\text{Im}(s)\in [T, 2T]$ is asymptotic to
$K_0 T G(T)/\sqrt{\log G(T)}$ uniformly in the range $ \log \log T \leq
G(T)\leq (\log T)^{\nu}$, where $K_0$ is a certain positive constant that
depends on $J$ and the $L_j$'s. This establishes a generalization of a
conjecture of Hejhal in this range. Moreover, the exponent $\nu$ verifies
$\nu\asymp 1/J$ as $J$ grows.
