  We show that a class of Dirichlet series ${\mathfrak{A}}^{\#}$ that is much
larger than the extended Selberg class ${\mathscr{S}}^{\#}$, and also contains
the standard as well as the tensor product, exterior square and symmetric
square $L$-functions of automorphic $L$-functions of $GL_n$ over number fields,
does not have any elements of degrees between $1$ and $2$. The proof of our
more general theorem is very different from the proof of Kaczorowski and
Perelli for the class ${\mathscr{S}}^{\#}$, and is much shorter and simpler
even in that case.
