Properties

Degree 1
Conductor 2011
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (2010, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5869725302$
$L(\frac12,\chi)$  $\approx$  $0.5869725302$
$L(\chi,1)$  $\approx$  0.4903903070
$L(1,\chi)$  $\approx$  0.4903903070

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.44167710532034174673713206960, −18.795199440490354601649902051234, −18.27412759337075320503104985715, −17.490840184183667435076921565564, −17.08966376820287751735699079595, −16.21920759150620512463129143550, −15.75251643168943629092111822102, −15.048096339780971782741167593233, −13.57760509429361157884363322358, −12.96453042401194124707524587451, −12.48131358584505874744725395229, −11.1967591835191728076773441134, −10.7334603597687525289307295638, −10.23049679982057810045994063912, −9.32601922242884721055551251865, −8.8188924322186848043221328198, −7.62830611616709875272880339443, −6.696374284341075413466670792813, −6.246244612512553351879993066, −5.66240680578143094399159035058, −4.58489182327237320470412077272, −3.2400073706179842695957863972, −2.32559914715180992040795361510, −1.42972080304135414620158102274, −0.390932760648706106866736217, 0.390932760648706106866736217, 1.42972080304135414620158102274, 2.32559914715180992040795361510, 3.2400073706179842695957863972, 4.58489182327237320470412077272, 5.66240680578143094399159035058, 6.246244612512553351879993066, 6.696374284341075413466670792813, 7.62830611616709875272880339443, 8.8188924322186848043221328198, 9.32601922242884721055551251865, 10.23049679982057810045994063912, 10.7334603597687525289307295638, 11.1967591835191728076773441134, 12.48131358584505874744725395229, 12.96453042401194124707524587451, 13.57760509429361157884363322358, 15.048096339780971782741167593233, 15.75251643168943629092111822102, 16.21920759150620512463129143550, 17.08966376820287751735699079595, 17.490840184183667435076921565564, 18.27412759337075320503104985715, 18.795199440490354601649902051234, 19.44167710532034174673713206960

Graph of the $Z$-function along the critical line