L${L}$‐functions of Kloosterman sheaves

Abstract In this article, we study a family of motives associated with the symmetric power of Kloosterman sheaves constructed by Fresán, Sabbah, and Yu. They demonstrated that for , the ‐functions of extend meromorphically to and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the ‐functions of some of the motives , with , 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.