On a result of Bao-Qin Li concerning Dirichlet series and shared values

Abstract

Li (2011) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) satisfy the same functional equation with \(a(1)=1\) and \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), then \(L_{1}=L_{2}\). Later on, Gonek--Haan--Ki (2014) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c)=L_2^{-1}(c)\) for a finite non-zero complex number \(c\), then \(L_1=L_2\). This implies that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), then \(L_1=L_2\). In this paper, we prove that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the zero degrees and satisfy \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), and if \(a_1(1)=a_2(1)\) or \(\lim_{r\rightarrow+\infty}\frac{T(r,L_2)}{T(r,L_1)}=1\), then \(L_{1}= L_{2}\). The main results obtained in this paper improve Theorem 1 from Li (2011) when the L-functions in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the zero degrees. Some examples are provided to show that the results obtained in this paper, in a sense, are best possible.