The mean value of symmetric square L-functions

The Mean-square of Dirichlet L-functions

where α and β are small complex numbers satisfying α, β ≪ 1/ log q. Ingham [Ing] considered an analogous moment for the Riemann zeta-function on the critical line with small shifts. Paley [Pal] considered the moment above for Dirichlet L-functions. Heath-Brown [HB] has computed a similar moment, but for all characters modulo q, in the case that α = β = 0. His result is Theorem 1 (HB). There are...

On Non-vanishing of Symmetric Square L-functions

We find a lower bound in terms of N for the number of newforms of weight k and level N whose symmetric square L-functions are non-vanishing at a fixed point s0 with 1 2 < Re(s0) < 1 or s0 = 1 2 .

Averages of symmetric square L-functions, and applications

We exhibit a spectral identity involving L(s,Symf) for f on SL2. Perhaps contrary to expectations, we do not treat L(s,Symf) directly as a GL3 object. Rather, we take advantage of the coincidence that the standard L-function for SL2 is the symmetric square for a cuspform on GL2 restricted to SL2. [1] As SL2 = Sp2, the integral identities obtained from Sp2n × Sp2n ⊂ Sp4n produce standard L-funct...

Symmetric Square L-Functions and Shafarevich-Tate Groups

CONTENTS We use Zagier's method to compute the critical values of the 1. Introduction symmetric square L-functions of six cuspidal eigenforms of level 2. Calculating the Critical Values one with rational coefficients. According to the Bloch-Kato 3. Tables of Results conjecture, certain large primes dividing these critical values 4. An Observation must be the orders of elements in generalised Sh...

Liftings and Mean Value Theorems for Automorphic L-functions