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Properties

Label 1-129-129.14-r1-0-0
Degree $1$
Conductor $129$
Sign $0.122 + 0.992i$
Analytic cond. $13.8629$
Root an. cond. $13.8629$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (−0.5 − 0.866i)7-s + (0.900 + 0.433i)8-s + (0.0747 − 0.997i)10-s + (0.222 − 0.974i)11-s + (0.0747 + 0.997i)13-s + (0.988 + 0.149i)14-s + (−0.900 + 0.433i)16-s + (−0.826 − 0.563i)17-s + (0.955 + 0.294i)19-s + (0.733 + 0.680i)20-s + (0.623 + 0.781i)22-s + (0.733 + 0.680i)23-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (−0.5 − 0.866i)7-s + (0.900 + 0.433i)8-s + (0.0747 − 0.997i)10-s + (0.222 − 0.974i)11-s + (0.0747 + 0.997i)13-s + (0.988 + 0.149i)14-s + (−0.900 + 0.433i)16-s + (−0.826 − 0.563i)17-s + (0.955 + 0.294i)19-s + (0.733 + 0.680i)20-s + (0.623 + 0.781i)22-s + (0.733 + 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.122 + 0.992i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ 0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6341410001 + 0.5607594716i\)
\(L(\frac12)\) \(\approx\) \(0.6341410001 + 0.5607594716i\)
\(L(1)\) \(\approx\) \(0.6260250306 + 0.2401996809i\)
\(L(1)\) \(\approx\) \(0.6260250306 + 0.2401996809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (0.955 + 0.294i)T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (0.365 + 0.930i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.955 + 0.294i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−28.345469905287224925903097969263, −27.52702899378577383661950627559, −26.52886502761589283563460067384, −25.37673056357779446265484356297, −24.529772541151885066765840416333, −22.86986366619147736862420133325, −22.2844747406970865690148584276, −20.89910716641532099098005815286, −20.02551468147067526608932319210, −19.33021054382930803484582986406, −18.18100302473068941130809279725, −17.238499285821174722364891620088, −15.944539876618729123887543307186, −15.18560826927399741774951800568, −13.14625277089899284734985087220, −12.416614605868642034844393757356, −11.58616129229436216278272065573, −10.252127736026189994678224989153, −9.09229574184270345185801025300, −8.25837141588891020943198032173, −6.98750376348606769910502482163, −5.02820075554546875241588347767, −3.69855218059115670226425319316, −2.36057110897465265407724176262, −0.58304998400355663166118511427, 0.941536461526471939806272640919, 3.30439392270870639698595473112, 4.704149581064248250252148105148, 6.48469185424224931551132535297, 7.110921212976148750720496149054, 8.32308994361516430250076044962, 9.50836318038948792109363754895, 10.746288937955139411934043324716, 11.62141467702097539630729402374, 13.63207418444841963713903462192, 14.25898275764365241109376267953, 15.72882714210746475372704488201, 16.246291903494888362387509257152, 17.37818183269971986318046979448, 18.65863850005026974391639940895, 19.34027250340295758098353685279, 20.21506421660893261791915121937, 21.9870060986449178841341579127, 23.07459146773085753851811005630, 23.72999060750395575970036183332, 24.74649885781009402877967824649, 26.01107573977281332019456436512, 26.901908242172794359024646962, 27.11204361950094080822093386878, 28.74388948738214357830837500246

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