L-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F(p) give rise to a lisse (Q(l)) over bar -sheaf Kl(n+1) on G(m,Fp) = P(Fp)(1) - {0, infinity}, which we call the Kloosterman sheaf. Let L(p)(G(m,Fp), Sym(k)Kl(n+1), s) be the L-function of the k-fold symmetric product of Kl(n+1). We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L(p)(G(m,Fp), Sym(k)Kl(n+1), s). We also prove similar results for circle times(k)Kl(n+1) and Lambda(k) Kl(n+1).