Geodesic
On the mean value of the generalized Dirichlet $L$-functions
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 597-620.

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Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac {\chi (n)}{(n+a)^s}, $$ where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
Classification : 11M20
Keywords: generalized Dirichlet $L$-functions; mean value properties; functional equation; asymptotic formula 
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     title = {On the mean value of the generalized {Dirichlet} $L$-functions},
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Ma, Rong; Yi, Yuan; Zhang, Yulong. On the mean value of the generalized Dirichlet $L$-functions. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 597-620. http://geodesic.mathdoc.fr/item/CMJ_2010__60_3_a1/