ar X iv : m at h / 05 05 39 9 v 1 [ m at h . N T ] 1 9 M ay 2 00 5 Uniform bound for Hecke L - functions

Most of arithmetically significant Dirichlet series such as the Riemann zeta-function ζ(s), Dirichlet L-functions, and Hecke L-functions associated with various cusp forms satisfy Riemannian functional equations connecting values at s = σ + it and 1 − s of respective functions. Essentially best possible estimates for these functions near the line σ = 1 and σ = 0 can usually be deduced from the definition of respective functions and their functional equations. From this, bounds in the critical strip 0 < σ < 1, in particular on the critical line σ = 1 2 , follow readily via the Phragmén–Lindelöf convexity principle; thus they are called convexity bounds. In general, there is a quantity B(g, t) characterising the size of a function g ( 1 2 + it ) of the above kind in a given t-range in such a way that the convexity bound is stated as