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Properties

Label 1-837-837.274-r1-0-0
Degree $1$
Conductor $837$
Sign $0.982 - 0.185i$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + 10-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.766 + 0.642i)20-s + (−0.766 + 0.642i)22-s + (−0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + 10-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.766 + 0.642i)20-s + (−0.766 + 0.642i)22-s + (−0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.878475344 - 0.2695909818i\)
\(L(\frac12)\) \(\approx\) \(2.878475344 - 0.2695909818i\)
\(L(1)\) \(\approx\) \(1.555086153 + 0.3730941737i\)
\(L(1)\) \(\approx\) \(1.555086153 + 0.3730941737i\)

Euler product

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

−21.87992769082968266192155876746, −21.461884476770746491077572405485, −20.49469047706648970752142129604, −19.54018578780197214214133231243, −18.93184186050937806080559309604, −18.34315384620410717236608072779, −17.0630799896883019696554114043, −16.28294503948784330211632892333, −15.36798413221160237691295812728, −14.34676736817910509776589536193, −13.7825231294690101139835903604, −13.21888355663264800051764236230, −12.25383831801571446845060670372, −11.26857947571718187385903877260, −10.61620347807485084776603685619, −9.69441562849129352311182526483, −9.236686794179522118148452809719, −7.55378561309140135516430992526, −6.50515316684995432718045264286, −5.98653190995471614640661136969, −5.07590813206300269162662215031, −3.59790062440740104821830764044, −3.28087297303390291522367149777, −2.08672831288153007551288677599, −1.06543337712761851347826153712, 0.50473639873957173677096771772, 2.23631482256186238694368049465, 2.97816539480836166594405000432, 4.17160779621293825876196935430, 5.2826832107625757996191053036, 5.64192293316540518388989702041, 6.743759696395277432461200310352, 7.54141924960713158704304358078, 8.5690118845082488926457031008, 9.571359320462943416586005278350, 10.12388340549855466911350754716, 11.709095806866583335773736179249, 12.47388089950874821480683105398, 13.012755299654691674772232243327, 13.701800812959304032739386187743, 14.70548361358540306670605838986, 15.46079627413909257425804070342, 16.32022350333696063570400052850, 16.80097128343159043807114330791, 17.90206588112483509476202899045, 18.32809901531678678205190190487, 19.954276187582581383009738551715, 20.42165063793840795282327039396, 21.20209851435841843850244164869, 22.1901048459806443053817494336

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