On the structure of the Selberg class, VI: non-linear twists

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<d<2, obtained by the same authors in Annals of Math. 173 (2011), 1397-1441.