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Properties

Label 1-837-837.560-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.827 + 0.561i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.961 + 0.275i)11-s + (0.848 − 0.529i)13-s + (0.241 + 0.970i)14-s + (−0.615 + 0.788i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.719 + 0.694i)20-s + (0.961 + 0.275i)22-s + (0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.961 + 0.275i)11-s + (0.848 − 0.529i)13-s + (0.241 + 0.970i)14-s + (−0.615 + 0.788i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.719 + 0.694i)20-s + (0.961 + 0.275i)22-s + (0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.827 + 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1416623581 - 0.4607325693i\)
\(L(\frac12)\) \(\approx\) \(-0.1416623581 - 0.4607325693i\)
\(L(1)\) \(\approx\) \(0.6194567917 - 0.3016145879i\)
\(L(1)\) \(\approx\) \(0.6194567917 - 0.3016145879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.848 - 0.529i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 + (-0.961 + 0.275i)T \)
13 \( 1 + (0.848 - 0.529i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (0.0348 - 0.999i)T \)
47 \( 1 + (-0.990 - 0.139i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.0348 - 0.999i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (-0.848 - 0.529i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−22.59388952611087974841701162908, −21.42203780608135675696668777261, −20.91188807807503019008185507284, −19.81666028904838370160585712019, −18.92310772184265825878127376456, −18.22653263352932660566851768507, −17.959641624482693021092716538303, −16.689881719446806141612721506, −16.11160517698826770947935532307, −15.404262910750873914944926957875, −14.474230468019776568390783310093, −13.562078262944017812351672032306, −12.90710553372449751910572859422, −11.43632329201355008004439420460, −10.8301905961085288168567971966, −9.79395412804649275595886743809, −9.27919224841762212373934479825, −8.49395201813635916225566566768, −7.34957803319304883988131961779, −6.49992541069479496678016808020, −5.78847403221990798637514851936, −5.115605976353341844303069708035, −3.27473725936345449319993813231, −2.34258885830354212045112891608, −1.378169423813844059518967307504, 0.149288056717818598260868503788, 1.088322634045934266925465601770, 2.25910690148625954358532364147, 3.08800039764611176456034004018, 4.22034267137650415892280714599, 5.45743115856047487552307212305, 6.560922398706313643243674354681, 7.28284834704931257923671821277, 8.44307875808103136720449444712, 9.1036935275365714105488760644, 10.073217284498350255106697175722, 10.51488737256963940517755960497, 11.36855736714611958913363419758, 12.79505830515381487129279044185, 13.07575175724356207168763988337, 13.768951775055047487376308423056, 15.3897174427254788104833489610, 15.981971120397211404627135478764, 16.94135255736552660986223645523, 17.4922743267200412665340354225, 18.2605927053865503028761070765, 18.97563159015404094270349475869, 20.107312318747297508392981915671, 20.49150986519745264774961114686, 21.17526564813890260967798323523

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