L-functions of symmetric powers of the generalized Airy family of exponential sums

TL;DR

This paper investigates the L-functions associated with symmetric powers of a specific family of exponential sums, providing explicit calculations of their degrees and local factors at infinity.

Contribution

It offers the first explicit computation of the degree and local factors of L-functions for symmetric powers of the generalized Airy exponential sums.

Findings

Computed the degree of the L-function as a rational function.

Determined the local factors at infinity for these L-functions.

Provided explicit formulas for the symmetric powers' L-functions.

Abstract

This paper looks at the L-function of the k-th symmetric power of the ell-adic sheaf over the affine line associated to the generalized Airy family of exponential sums. We compute the degree of this rational function and the local factors at infinity.