The completed standard \(L\)-function of modular forms on \(G_2\) (Q2163314)

In the paper under the review, the authors study standard \(L\)-functions for certain cuspidal automorphic representations of the split exceptional linear algebraic group over \(\mathbb{Q}\) of Dynkin type \(G_2\). Suppose that \(\varphi\) is a level one cuspidal modular form on \(G_2\) of positive weight \(l\) which generates the cuspidal automorphic representation \(\pi\). Then \(\pi\) is of the form \(\pi = \pi_f \otimes \pi_{\infty}\), where \(\pi_f\) is unramified at every finite place, and to \(\pi_{\infty}\) one can attach a non-generic discrete series \(\pi_{l, \infty}\) of \(G_2(\mathbb{R})\). The authors define the archimedean \(L\)-factor \(L_{\infty}(\pi_{l, \infty}, s)\) and the completed \(L\)-function \(\Lambda(\pi, \mathrm{Std}, s) = L_{\infty}(\pi_{l, \infty}, s) L(\pi, \mathrm{Std}, s)\). Under a certain assumption on the Fourier coefficients of \(\varphi\), authors prove that \(\Lambda(\pi, \mathrm{Std}, s) = \Lambda(\pi, \mathrm{Std}, 1-s)\). Also, a Dirichlet series for \(L(\pi, \mathrm{Std}, s)\) is obtained and it is proven that \(L(\pi, \mathrm{Std}, s)\) vanishes to order \(3\) at negative even integers of sufficiently large absolute value and vanishes to order \(4\) at negative odd integers of sufficiently large absolute value.