Properties

Degree 1
Conductor $ 3^{3} $
Sign $0.686 - 0.727i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.173 − 0.984i)23-s + ⋯
L(s,χ)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.686 - 0.727i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.686 - 0.727i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.686 - 0.727i$
motivic weight  =  \(0\)
character  :  $\chi_{27} (23, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 27,\ (1:\ ),\ 0.686 - 0.727i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.049901826 - 0.4528834405i$
$L(\frac12,\chi)$  $\approx$  $1.049901826 - 0.4528834405i$
$L(\chi,1)$  $\approx$  0.9010756389 - 0.2697643375i
$L(1,\chi)$  $\approx$  0.9010756389 - 0.2697643375i

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

−37.53366505966979641594970629121, −36.31601986524997072485111624701, −35.271037995674476437368408227463, −33.91780358148320433489699433179, −32.98674963355186940877977554391, −31.718831195662361340027856950063, −29.77775734851037840073813275153, −28.414887900591098520993224355502, −27.6552187116060785709295681227, −25.81026275698904349409049615293, −25.13773893512662082183113132887, −23.92546177783779570996477147733, −22.09396292650039540476286166698, −20.58807361590179568821240527266, −18.911677784827641022457791120299, −17.78030900596240395762260934212, −16.60375858648333179035323413769, −15.115001492003774955207310958172, −13.711768454431918706781489429493, −11.58392197475339631659235490554, −9.61628856160981720152308141603, −8.74806514964328799070173717152, −6.65018619049284388477146828262, −5.286877330186552482557673813398, −1.75769826820186337067354527515, 1.441436999269837861042254203, 3.64238602984040811578385626578, 6.46316713135961146638009148476, 8.30202333589498240547601198673, 9.98592800196832013395963330654, 10.949752307846942382238008247631, 12.846732643381331047996593699754, 14.27180222905270437106694934159, 16.64245809243140613011981334943, 17.53988099444550283472867749350, 18.894953972888724681366390607190, 20.34214067321896580962533682577, 21.41868002013041392172732737095, 22.81266195043175413776378616958, 24.90803354656048745606169698801, 26.035668549456502540633148159668, 27.16839185096248515389794192688, 28.489658163077822255487108093517, 29.95148265347417789192791166868, 30.25607698745007339098413774007, 32.46834799858984316305419471906, 33.70052728983286179445406181791, 35.11178690130733074172521844083, 36.358351591028983047387612389514, 37.33040604432932625571887814629

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