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Properties

Label 1-3e3-27.20-r1-0-0
Degree $1$
Conductor $27$
Sign $0.686 + 0.727i$
Analytic cond. $2.90155$
Root an. cond. $2.90155$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(2.90155\)
Root analytic conductor: \(2.90155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (1:\ ),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049901826 + 0.4528834405i\)
\(L(\frac12)\) \(\approx\) \(1.049901826 + 0.4528834405i\)
\(L(1)\) \(\approx\) \(0.9010756389 + 0.2697643375i\)
\(L(1)\) \(\approx\) \(0.9010756389 + 0.2697643375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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