Properties

Degree 1
Conductor $ 3^{2} $
Sign $0.642 + 0.766i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s + 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.642 + 0.766i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.642 + 0.766i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $0.642 + 0.766i$
motivic weight  =  \(0\)
character  :  $\chi_{9} (2, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 9,\ (1:\ ),\ 0.642 + 0.766i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.113946943 + 0.5194419906i$
$L(\frac12,\chi)$  $\approx$  $1.113946943 + 0.5194419906i$
$L(\chi,1)$  $\approx$  1.136275918 + 0.4135706123i
$L(1,\chi)$  $\approx$  1.136275918 + 0.4135706123i

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

−47.55248127455483201705852602745, −45.95935080542520981794114449963, −44.781583927578735998719033134104, −42.19507464737363382501335827663, −41.265221376972713038042216016559, −39.68758273933563024241709715528, −38.02702866170852311461138362749, −37.27484517767191228753209201241, −34.97343033685754567017430433836, −33.04716874508529738151387747530, −31.55515371295916543880032766067, −29.99559371181156904706964137279, −28.821574701722030960999337413674, −26.9541568022766289467089005514, −24.80445956457406238489997737585, −22.55618331766574485452776535305, −21.6799089672262001832976919749, −19.55064984717247446010842616849, −18.1206180739956670326074406043, −15.105372647488006472027705459194, −13.38637373763353051188582047267, −11.407478947536363718469485584758, −9.56544293453670367607222319631, −5.911589958627904376506207333864, −2.90199460773728348151070380596, 4.57573576242485587483690886648, 6.89180300406244509022667855056, 9.262408964681987903208968444344, 12.586494690151173034212579462720, 14.08927896436677198005366627039, 16.2551781515321363983903806222, 17.47183618509653816175409815785, 20.29865916653771873113403507250, 22.1765179023783446952084486311, 23.83477567755688274260462694769, 25.21569578513310088903785578470, 26.763428341842772681883211365481, 28.9577216343121731590172682939, 30.90230473712494219306650729593, 32.573271591370362790797457866918, 33.412433807129446974963822755490, 35.48489755956410935564836049716, 36.48187828065801756344300334031, 39.07364690643271231694332487226, 40.3943285623758754227281525746, 41.693019157648402269033143994997, 43.28558880343466646432456708351, 44.35691248613978483473807819478, 45.96161983593313488222891071970, 48.3575107148706247398265274126

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