  First we show that the abscissae of uniform and absolute convergence of
Dirichlet series coincide in the case of $L$-functions from the Selberg class
$\mathcal{S}$. We also study the latter abscissa inside the extended Selberg
class, indicating a different behavior in the two classes. Next we address two
questions about majorants of functions in $\mathcal{S}$, showing links with the
distribution of the zeros and with independence results.
