Infinite symmetric power L -functions of the hyper-Kloosterman family

The infinite symmetric power L-functions play a fundamental role in Wan’s groundbreaking work on Dwork’s conjecture. Building upon this foundation, Haessig established p-adic estimates for these L-functions in the case of the one-dimensional Kloosterman family. In this paper, we extend Haessig’s results by deriving a uniform lower bound for the q-adic Newton polygons of the infinite symmetric power L-functions associated with the hyper-Kloosterman family.