Functional Equations of $L$-Functions for Symmetric Products of the Kloosterman Sheaf

We determine the (arithmetic) local monodromy at 0 and at $\mathrm{\infty }$ of the Kloosterman sheaf using local Fourier transformations and Laumon’s stationary phase principle. We then calculate $ϵ$-factors for symmetric products of the Kloosterman sheaf. Using Laumon’s product formula, we get functional equations of $L$-functions for these symmetric products, and prove a conjecture of Evans on signs of constants of functional equations.