ContentsNotation 1. Introduction 2. Outline of the method 3. The case 0~l, F(s) is an absolutely convergent Dirichlet series a(n) F(s)= E n ~ n=l (ii) (Analytic continuation) For some integer m >~ O, (s-1)mF(s) is an entire function of finite order. (iii) (Functional equation) F(s) satisfies a functional equation of the form &(8) =~c~(1-8) The first author was partially supported by the CNR Visiting Professors Program and the KBN Grant 2 P03A 02809. 208 J. KACZOROWSKI AND A. PERELLI where r 9 (s) = Qs YI r(Ass+m)F(s) j=t with Q>0, Aj>0, Repj~>0 and [021=1. (iv) ( Ramanujan hypothesis) For every r a( n ) << n ~ . (v) (Euler product) For a sufficiently large, b(n) log F(s)= Z ns n=l where b(n)=0 unless n is a positive power of a prime, and b(n)<

The Selberg class S 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 S. 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 S. 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 S.

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We continue our investigations of the analytic properties of nonlinear twists of L-functions developed in [4], [5] and [7]. Let F (s) be an L-function of degree d. First we extend the transformation formula in [5], relating a twist F (s; f ) with leading exponent κ 0 > 1/d to its dual twist F (s; f * ). Then we combine the results in [7] with such a transformation formula to obtain the analytic properties of new classes of nonlinear twists. This allows to detect several new cases of resonance of the classical L-functions. (2000): 11 M 41 Mathematics Subject Classification

We obtain the analytic properties of the standards twists of the L-functions in the Selberg class, i.e. meromorphic continuation, polar structure and polynomial growth on vertical lines. We also obtain uniform bounds on vertical strips. Moreover, as an application we improve certain estimates for exponential sums involving Fourier coefficients of modular forms obtained by Iwaniec-Luo-Sarnak. The results in this paper are important for the proof of the degree conjecture, when 1