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

Abstract 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.