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Properties

Label 1-3e2-9.2-r1-0-0
Degree $1$
Conductor $9$
Sign $0.642 + 0.766i$
Analytic cond. $0.967184$
Root an. cond. $0.967184$
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  + (0.5 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.967184\)
Root analytic conductor: \(0.967184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 9,\ (1:\ ),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113946943 + 0.5194419906i\)
\(L(\frac12)\) \(\approx\) \(1.113946943 + 0.5194419906i\)
\(L(1)\) \(\approx\) \(1.136275918 + 0.4135706123i\)
\(L(1)\) \(\approx\) \(1.136275918 + 0.4135706123i\)

Euler product

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

−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