Zero-Free Region For L-Functions

The development of modern number theory is motivated by the Langlands program. According to the program each L-function can be expressed as a product of L-functions of automorphic cuspidal representation of GL_m, m ≥ 1. Therefore, it is important to investigate analytic properties of automorphic L-functions.In this paper, we study zero-free region of L-functions of new forms for the Hecke congruence group. Let q be a square-free positive integer and k an even natural number. Let f be a new form of weight k for the Hecke congruence group Γ₀(q)- Letbe its normalized Fourier expansion at the cusp ∝. Then is an L-function of degree 2 with conductor q. The GRH predicts that all the nontrivial zeros of L(s, f) in the critical strip lie on the critical line (?)_s = 1/2.We begin with describing the method in general terms. Let ρ_f=1/2+ir f denote a nontrivial zero of L(s,f). Then the GRH predicts that r_f .To investigate those nontrivial zeros, we definewhere R > 1 is a parameter, and φ(x) is a test function of Schwartz class with compact support in (-v, v). Letwhere h(x) is a smooth function supported in [0, +∞) and K a parameter. In our discussions, we will takeAlso we will make the following hypothesis: Hypothesis*: Let α > 0 and 0 < δ < 3/4. Thenwhere the implied constant depends on 5 only.Our main result is as follows.Theorem 1. .Assume Hypothesis* and assume further that all the zeros of L(s,f) forf ∈ Hk(q) are real or lie on the critical line. ThenL(s, f) ≠ 0, for s > max ,provided that k is sufficiently large in terms of ε.Our result generalizes the results of H. Iwaniec, W. Luo and P. Sarnak (see [1]). They are the first to have noticed the fact that a zero-free region of L-functions for cusp forms with respect to the modular group 5L₂(Z) can be derived from an estimate about a certain exponential sum over primes in arithmetic progression, and have obtained the zero-free region s > 10/11 + ε. It is natural to consider the more general problem concerning .^-functions of new forms with respect to the Hecke congruence group. Unfortunately, our results is slightly worse than that in [1] due to the crude estimate for a key term in A(K,φ). In our proof, we find that the test function denned by (1) satisfies φ|^(x)x < 1, where φ|^(x) is the Fourier transform of φ. Using this fact, the proof in [1] can be simplified slightly.