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Properties

Label 1-4029-4029.1010-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.389 - 0.920i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.297 + 0.954i)2-s + (−0.822 + 0.568i)4-s + (−0.685 − 0.728i)5-s + (−0.491 + 0.871i)7-s + (−0.787 − 0.616i)8-s + (0.491 − 0.871i)10-s + (−0.728 − 0.685i)11-s + (0.822 − 0.568i)13-s + (−0.977 − 0.209i)14-s + (0.354 − 0.935i)16-s + (0.0603 + 0.998i)19-s + (0.977 + 0.209i)20-s + (0.437 − 0.899i)22-s + (0.382 − 0.923i)23-s + (−0.0603 + 0.998i)25-s + (0.787 + 0.616i)26-s + ⋯
L(s)  = 1  + (0.297 + 0.954i)2-s + (−0.822 + 0.568i)4-s + (−0.685 − 0.728i)5-s + (−0.491 + 0.871i)7-s + (−0.787 − 0.616i)8-s + (0.491 − 0.871i)10-s + (−0.728 − 0.685i)11-s + (0.822 − 0.568i)13-s + (−0.977 − 0.209i)14-s + (0.354 − 0.935i)16-s + (0.0603 + 0.998i)19-s + (0.977 + 0.209i)20-s + (0.437 − 0.899i)22-s + (0.382 − 0.923i)23-s + (−0.0603 + 0.998i)25-s + (0.787 + 0.616i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.389 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3827295085 - 0.2535765010i\)
\(L(\frac12)\) \(\approx\) \(0.3827295085 - 0.2535765010i\)
\(L(1)\) \(\approx\) \(0.7422695159 + 0.3040236329i\)
\(L(1)\) \(\approx\) \(0.7422695159 + 0.3040236329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.297 + 0.954i)T \)
5 \( 1 + (-0.685 - 0.728i)T \)
7 \( 1 + (-0.491 + 0.871i)T \)
11 \( 1 + (-0.728 - 0.685i)T \)
13 \( 1 + (0.822 - 0.568i)T \)
19 \( 1 + (0.0603 + 0.998i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.805 - 0.592i)T \)
31 \( 1 + (-0.0904 + 0.995i)T \)
37 \( 1 + (-0.326 + 0.945i)T \)
41 \( 1 + (0.0302 - 0.999i)T \)
43 \( 1 + (0.911 + 0.410i)T \)
47 \( 1 + (-0.663 - 0.748i)T \)
53 \( 1 + (-0.616 + 0.787i)T \)
59 \( 1 + (0.180 + 0.983i)T \)
61 \( 1 + (-0.437 - 0.899i)T \)
67 \( 1 + (-0.885 + 0.464i)T \)
71 \( 1 + (-0.871 + 0.491i)T \)
73 \( 1 + (-0.209 + 0.977i)T \)
83 \( 1 + (0.180 - 0.983i)T \)
89 \( 1 + (-0.992 - 0.120i)T \)
97 \( 1 + (-0.945 - 0.326i)T \)
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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

−18.79748180189169876275458320071, −17.922196750717326377062292058422, −17.672329327683681788602744153255, −16.45458824900460048918055351668, −15.74182506489496056465579438022, −15.150524749937567643075103487143, −14.346209907066151366867285970344, −13.67591114020833918045552200094, −13.12866691131238242399586954928, −12.45885432215641046573127878584, −11.5597389173769571133222755774, −11.00109484656985998641279568005, −10.58999533273981333116904482958, −9.740818406356452484468589186893, −9.16991119382470902465988646032, −8.13009896870750737322555400885, −7.374402421875393598041981466535, −6.69864371305279274073649391663, −5.86817594363656245010546842116, −4.73170110357943589171561257254, −4.26739400244363032113141668877, −3.41180632123929797048536220757, −2.929578991715548081668146358907, −1.97001292930776563605361818464, −0.93943800200767256525941446564, 0.14853334495482823252105230792, 1.23497267635383907431484152781, 2.81762115504125059439757214115, 3.32519408134351826118288034350, 4.21085684754119018173453448776, 5.01261840128292076897449211665, 5.70678845194483201306814117686, 6.16824498798038605325081802829, 7.13626400283497715655738210579, 8.04494666543313949818269692172, 8.50809053598451166804096881541, 8.85992400977840003059088826329, 9.91614272198239478915893653879, 10.72600177883331554954959848165, 11.80686319931827893497124942136, 12.36576467191935162300889441215, 12.91235141274612669365512046020, 13.52773224460332955008611251292, 14.34381514919777192926153299599, 15.20447297224703553739638620157, 15.765255221979519151225961434116, 16.100530683545825391044935229898, 16.69919438321732773934815036989, 17.55797838241866568235316588007, 18.403494643483562551809886695016

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