The Selberg class is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of . Such a classification is based on a real-valued invariant d called degree, and the degree conjecture asserts that d∈ℕ for every L-function in . The degree conjecture has been proved for d<5/3, and in this paper we extend its validity to d<2. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the L-functions in .