L-functions of Kloosterman sheaves

In this article, we study a family of motives M-n+1(k) associated with the symmetric power of Kloosterman sheaves constructed by Fres & aacute;n, Sabbah, and Yu. They demonstrated that for n = 1, the L-functions of M-2(k) extend meromorphically to C and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the L-functions of some of the motives M-n+1(k), with n > 1, as well as other related two-dimensional motives. In particular, we prove several conjectures of Evans type, which relate moments of Kloosterman sheaves and Fourier coefficients of modular forms.