Testing the functional equation of a high-degree Euler product

Abstract

We study the L-functions associated to Siegel modular forms -equivalently, automorphic representations of GSp(4, ‫ށ‬ ‫ޑ‬ ) -both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation for an L-function, and we apply this to the degree-10 adjoint L-function associated to a Siegel modular form.

Maass Forms on GL(3) and GL(4)

“…If that is true, then we can use the functional equation data and Dirichlet coefficients to compute the standard L−function, which has degree 5. (That the standard L−function can be reconstructed from the spin L−function can be seen from an examination of the explicit representations given in [13].) We computed the degree 5 L−functions in several cases and found that they satisfy the expected functional equation to high accuracy.…”

“…That is, we used the functional equation and Dirichlet coefficients of the degree 3 L−function to determine the functional equation and Dirichlet coefficients of the degree 6 L−function. We then used the method in Theorem 4.2 of [13] to verify that the L−function satisfies the conjectured functional equation. In all cases we find that degree 6 L−function satisfies the expected functional equation.…”

“…The terms decrease rapidly: the coefficients of b 1 (15) and b 2 (15) are less than 2 • 10 −13 . The exponential decay in the contributions of the Dirichlet coefficients is a general phenomenon [4,13]. With the same s 0 , now choose g 2 (s) = e −is/2 .…”

Numerical computation of a certain Dirichlet series attached to Siegel modular forms of degree two

“…Another way we verify that the Gram data is correct is we check, as in [6], that the series D F G for F and G not in spezialschaar, satisfies the functional equation shown in [9]. 3.6.…”

Evaluating -functions with few known coefficients

“…The degree 4, 5, and 10 L-functions are called, respectively, the spinor, standard, and adjoint, and are denoted L(s, F, spin), L(s, F, stan), and L(s, F, adj). Proposition 2.1, taken from [9], summarizes the properties of those L-functions for a weight k Siegel modular form on Sp(4, Z).…”