Beyond the extended Selberg class: $d_F\le 1$

We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\mathfrak{A}^{\#}$ contains both the extended Selberg class $\mathscr{S}^{\#}$ of Kaczorowski and Perelli as well as many $L$-functions attached to automorphic representations of ${\rm GL}_n({\mathbb A}_K)$, where ${\mathbb A}_K$ denotes the ad\`eles over the number field $K$ (these representations need not be unitary or generic). This is in contrast to the class $\mathscr{S}^{\#}$ which is smaller and is known to contain, very few of these $L$-functions. The larger class is obtained by weakening the requirement for absolute convergence, allowing a finite number of poles, allowing more general gamma factors and by allowing the series to have trivial zeros to the right of $\mathrm{Re}(s)=1/2$, while retaining the other axioms of the extended Selberg class. We will classify series in $\mathfrak{A}^{\#}$ of degree $d$ when $d\le 1$ (when $d=1$, we will assume absolute convergence in $\mathrm{Re}(s)>1$). We will further prove a primitivity result for the $L$-functions of cuspidal eigenforms on ${\rm GL}_2({\mathbb A}_{\mathbb Q})$ and a theorem allowing us to compare the zeros of tensor product $L$-functions of ${\rm GL}_n({\mathbb A}_K)$ which cannot be deduced from previous classification results. The second class $\mathfrak{G}^{\#}\subset\mathfrak{A}^{\#}$, which also contains $\mathscr{S}^{\#}$, more closely models the behaviour of $L$-functions of unitary globally generic representations of ${\rm GL}_n({\mathbb A}_K)$.