$L$-functions of twisted diagonal exponential sums over finite fields

Let ${\bf F}_q$ be the finite field of $q$ elements with characteristic $p$ and ${\bf F}_{q^m}$ its extension of degree $m$. Fix a nontrivial additive character $\Psi$ and let $\chi _1,..., \chi _n$ be multiplicative characters of ${\bf F}_p.$ For

$f(x_1,...,x_n) \in {\bf F}_q[x_1,x_1^{-1},...,x_n,x^{-1}_n],$

one can form the twisted exponential sum $S^*_m(\chi _1,...,\chi _n,f)$. The corresponding $L$-function is defined by

$L^*(\chi _1,..., \chi _n,f;t)=\hbox {exp}(\sum^{\infty}_{m=0}S^*_m(\chi _1,...,\chi _n, f){\frac {t^m} {m}} ).$

In this paper, by using the $p$-adic gamma function and the Gross-Koblitz formula on Gauss sums, we give an explicit formula for the $L$-function $L^*(\chi _1,...,\chi _n, f;t)$ if $f$ is a Laurent diagonal polynomial. We also determine its $p$-adic Newton polygon.