Some remarks on the convergence of the Dirichlet series of L-functions and related questions

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 $\S$. 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 $\S$, showing links with the distribution of the zeros and with independence results.