Symmetric power L-functions of a weighted hyper-Kloosterman family

Abstract: As a natural generalization of the classical hyper-Kloosterman family studied by D. Haessig and S. Sperber, we study the $k$-th symmetric power $L$-functions attached to a weighted hyper-Kloosterman family $$Kl_{n,m}(t;x_{1},\cdot\cdot\cdot,x_{n})=x_{1}^{{m}+x_{2}\cdot\cdot\cdot+x_{n}+\frac{t}{x_{1}x_{2}\cdot\cdot\cdot} x_{n}}.$$ Under suitable conditions, we determine the bounds of degrees of these $L$-functions and prove that their $q$-adic Newton polygons admit uniform lower bounds.