119 Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a0 + a1x + ... + akxk is a k-degree polynomial with integral coefficients such that (p, a0, a1, ..., ak) = 1, for any integer m, we study the asymptotic property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}} {p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } ,$$\end{document} where e(y) = e2πiy. The main purpose is to use the analytic method to study the 2m-th power mean of Dirichlet L-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results. weight asymptotic formula modulo p trigonometric sums method asymptotic properties en coefficient sum articles 2021-11-01T18:14 function false primes formula degree polynomial polynomials properties general trigonometric sums main purpose A0 https://scigraph.springernature.com/explorer/license/ Akxk extension interesting asymptotic formula analytic methods 2009-09-01 411 Dirichlet L https://doi.org/10.1007/s12044-009-0046-8 Dirichlet character modulo p purpose integral coefficients article previous results a1x results power means character modulo p On the 2m-th power mean of Dirichlet L-functions with the weight of trigonometric sums means 2009-09 School of Science, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, People’s Republic of China School of Science, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, People’s Republic of China The School of Electronic and Information Engineering, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, People’s Republic of China The School of Electronic and Information Engineering, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, People’s Republic of China 10.1007/s12044-009-0046-8 doi dimensions_id pub.1031722255 Zhang Junhuai 4 Pure Mathematics Springer Nature Proceedings - Mathematical Sciences 0973-7685 0253-4142 Ma Rong Mathematical Sciences Zhang Yulong Springer Nature - SN SciGraph project