Motivic orthogonal two-dimensional representations of Gal(ℚ/ℚ)

Given a two-dimensional compatible family of ℓ-adic representations which is motivic and which respects an orthogonal form up to similitudes, we show how to express its L-function in terms of a Hecke character. We give several examples and in particular we analyze a representation associated to a certain K3 surface which arose in the study of Kloosterman sums.