To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that

1. $\mathcal S_0=\{1\}$ (Conrey-Ghosh, 1993);

2. $\mathcal S_d=\emptyset$ for $0<d<1$ (Richert, 1957 and others);

3. $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$-functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (Kaczorowski-Perelli, 1999);

4. $\mathcal S_d=\emptyset$ for $1<d<2$ (Kaczorowski-Perelli, 2002 and 2011).

Apart from these results, I think that nothing has been established in general. A nice survey of the results obtained so far can be found in the introduction to

J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1<d<2$", Ann. of Math. (2) 173 (2011), 1397-1441.

Note that, in fact, Kaczorowski and Perelli prove their results for functions in the so-called extended Selberg class $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property.

To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that

1. $\mathcal S_0=\{1\}$ (Conrey-Ghosh, 1993);

2. $\mathcal S_d=\emptyset$ for $0\lt d\lt1$ (Richert, 1957 and others);

3. $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$-functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (Kaczorowski-Perelli, 1999);

4. $\mathcal S_d=\emptyset$ for $1\lt d\lt2$ (Kaczorowski-Perelli, 2002 and 2011).

Apart from these results, I think that nothing has been established in general. A nice survey of the results obtained so far can be found in the introduction to

J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1\lt d\lt2$", Ann. of Math. (2) 173 (2011), 1397-1441.

Note that, in fact, Kaczorowski and Perelli prove their results for functions in the so-called extended Selberg class $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property.

To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that

• $\mathcal S_0=\{1\}$ (Conrey-Ghosh, 1993);
• $\mathcal S_d=\emptyset$ for $0<d<1$ (Richert, 1957 and others);
• $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$-functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (Kaczorowski-Perelli, 1999);
• $\mathcal S_d=\emptyset$ for $1<d<2$ (Kaczorowski-Perelli, 2002 and 2011).

Apart from these results, I think that nothing has been established in general. A nice isurvey of the results obtained so far can be found in the introduction to

J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1<d<2$", Ann. of Math. (2) 173 (2011), 1397-1441.

Note that, in fact, Kaczorowski and Perelli prove their results for functions in the so-called extended Selberg class $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property.

To the best of my knowledge, we are very far from proving such a conjecture. If $\mathcal S_d$ is the subclass of the Selberg class $\mathcal S$ consisting of the functions of degree $d$ then it is known that

1. $\mathcal S_0=\{1\}$ (Conrey-Ghosh, 1993);

2. $\mathcal S_d=\emptyset$ for $0<d<1$ (Richert, 1957 and others);

3. $\mathcal S_1$ consists of the Riemann zeta function $\zeta(s)$ and the shifted Dirichlet $L$-functions $L(s+i\tau,\chi)$ with $\tau\in\mathbb R$ and $\chi$ a primitive character (Kaczorowski-Perelli, 1999);

4. $\mathcal S_d=\emptyset$ for $1<d<2$ (Kaczorowski-Perelli, 2002 and 2011).

Apart from these results, I think that nothing has been established in general. A nice survey of the results obtained so far can be found in the introduction to

J. Kaczorowski, A. Perelli, "On the structure of the Selberg class, VII: $1<d<2$", Ann. of Math. (2) 173 (2011), 1397-1441.

Note that, in fact, Kaczorowski and Perelli prove their results for functions in the so-called extended Selberg class $\mathcal S^\sharp$, whose elements are not required to satisfy the Ramanujan hypothesis and the Euler product property.