Determining L-functions by twists

Hybrid Bounds for Twists of Gl(3) L-functions

Let π be a Hecke-Maass cusp form for SL(3,Z) and χ = χ1χ2 a Dirichlet character with χi primitive modulo Mi. Suppose that M1, M2 are primes such that max{(M |t|)1/3+2δ/3,M2/5|t|−9/20,M1/2+2δ|t|−3/4+2δ}(M |t|)ε < M1 < min{(M |t|)2/5, (M |t|)1/2−8δ}(M |t|)−ε for any ε > 0, where M = M1M2, |t| ≥ 1 and 0 < δ < 1/52. Then we have L ( 1 2 + it, π ⊗ χ ) ≪π,ε (M |t|).

Cubic Twists of Gl(2) Automorphic L-functions

Let K = Q(√−3) and let π be a cuspidal automorphic representation of GL(2, AK). Consider the family of twisted L-functions L(s, π⊗χ) where χ ranges over the cubic Hecke characters of K. In this paper the mean value of this family of L-functions is computed; the result is consistent with the generalized Lindelöf hypothesis. From this mean value result a nonvanishing theorem is established: for g...

The Second Moment of Quadratic Twists of Modular L-functions

The family of quadratic twists of a modular form has received much attention in recent years. Motivated by the Birch-Swinnerton-Dyer conjectures, we seek an understanding of the central values of the associated L-functions, and while this question has been investigated extensively, much remains unknown. One important theme in this area concerns the moments of these central L-values. Thanks to t...

Non-vanishing of Quadratic Twists of Modular L-functions

The Second Moment of Dirichlet Twists of Hecke L-functions