  We construct motives over the rational numbers associated with symmetric
power moments of Kloosterman sums, and prove that their L-functions extend
meromorphically to the complex plane and satisfy a functional equation
conjectured by Broadhurst and Roberts. Although the motives in question turn
out to be "classical", we compute their Hodge numbers by means of the irregular
Hodge filtration on their realizations as exponential mixed Hodge structures.
We show that all Hodge numbers are either zero or one, which implies potential
automorphy thanks to recent results of Patrikis and Taylor.
