Properties

Degree 1
Conductor 3
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s + 40-s − 41-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 3 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3} (2, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 3,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4808675576$
$L(\frac12,\chi)$  $\approx$  $0.4808675576$
$L(\chi,1)$  $\approx$  0.6045997880
$L(1,\chi)$  $\approx$  0.6045997880

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

−63.17699276733910829118311838811, −62.20607812228213648686964588013, −60.02685745671590223526149077217, −57.58405636044918378148658948822, −55.6425587002162213738564033666, −54.19384310155191357860186830353, −52.4967495990607536672882221717, −50.375138650636265982821917341449, −47.5141045101173221149363569968, −46.27411802351314008510487166014, −44.12057291207220244202440459309, −42.6163792261575675743228068514, −39.48520726092935076736629266003, −37.55179655636462701986044645770, −35.60841265393863465481291012803, −33.89738892725941901767787403300, −30.74504026138249573780824181050, −28.21816450623338609318302976031, −26.57786873577458531458435093753, −24.05941485649345077459305359321, −20.45577080774249285344502583131, −18.26199749569312756892441409359, −15.70461917672162556516555088043, −11.24920620777293524970502567886, −8.03973715568146668171362321417, 8.03973715568146668171362321417, 11.24920620777293524970502567886, 15.70461917672162556516555088043, 18.26199749569312756892441409359, 20.45577080774249285344502583131, 24.05941485649345077459305359321, 26.57786873577458531458435093753, 28.21816450623338609318302976031, 30.74504026138249573780824181050, 33.89738892725941901767787403300, 35.60841265393863465481291012803, 37.55179655636462701986044645770, 39.48520726092935076736629266003, 42.6163792261575675743228068514, 44.12057291207220244202440459309, 46.27411802351314008510487166014, 47.5141045101173221149363569968, 50.375138650636265982821917341449, 52.4967495990607536672882221717, 54.19384310155191357860186830353, 55.6425587002162213738564033666, 57.58405636044918378148658948822, 60.02685745671590223526149077217, 62.20607812228213648686964588013, 63.17699276733910829118311838811

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