Properties

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

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s + 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s − 37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s + 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 35-s − 37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.642 + 0.766i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.642 + 0.766i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(72\)    =    \(2^{3} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.642 + 0.766i$
motivic weight  =  \(0\)
character  :  $\chi_{72} (11, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 72,\ (0:\ ),\ 0.642 + 0.766i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8256783936 + 0.3850201581i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8256783936 + 0.3850201581i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9657295487 + 0.2418462325i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9657295487 + 0.2418462325i\)

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

−31.41750544205144077923310704548, −30.52325296682130191117812107755, −29.24257735426601907009727017624, −28.23440745442363445517025022235, −27.08266425690759396433972013943, −26.32332905636736910326283304543, −24.495186011995033430334416316712, −24.10088757754853338482249620263, −22.83902782531965293166734826815, −21.41205749308946722549639921677, −20.34071796558502850726197051049, −19.503557143583352681058482142154, −18.02081219647978144864204048689, −16.69240992689120755000267012648, −16.0230132207712621025606396335, −14.30662366215205834290305926085, −13.35613524784961168721824939651, −11.85602292923144602115842235229, −10.910787912465851977167363821390, −9.14548552589160229265556507650, −8.13183863401421268741523653060, −6.65778239132726212866136137028, −4.860245271433104475227951032356, −3.74002024390602897682147946262, −1.280152197260028030800112306148, 2.24004668425538474525233672394, 3.84186622036355904314648850343, 5.56009399288233496114948471525, 7.04610367991045475940697044530, 8.27934023819982130331178300235, 9.78287036852542958536656612719, 11.22443392675669114780611968392, 12.083209241227550718772095990492, 13.706457356116864517578608271846, 15.118598841467356039227958215117, 15.58456159252951795433077305233, 17.579108694749484804602030326419, 18.295376777617149728141500040726, 19.55003745177711351360128864969, 20.683856857880770865253802663443, 22.18109369445376156327064564481, 22.74636448527918262607168535777, 24.20369078163529232283781407910, 25.26505887346781658518871709209, 26.36871616010120181541707939936, 27.5523666212794504881485136308, 28.29821303154952302982613649817, 29.86383860276153121854188414741, 30.76198118182756399963417571918, 31.47365284387212510527484027178

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