  We determine the (arithmetic) local monodromy at 0 and at $\infty$ of the
Kloosterman sheaf using local Fourier transformations and Laumon's stationary
phase principle. We then calculate $\epsilon$-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.

Number Theory;Algebraic Geometry