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Properties

Label 1-837-837.113-r1-0-0
Degree $1$
Conductor $837$
Sign $0.0258 + 0.999i$
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  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.0348 − 0.999i)11-s + (0.559 + 0.829i)13-s + (−0.848 − 0.529i)14-s + (−0.241 − 0.970i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.882 + 0.469i)20-s + (−0.882 + 0.469i)22-s + (−0.0348 + 0.999i)23-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.0348 − 0.999i)11-s + (0.559 + 0.829i)13-s + (−0.848 − 0.529i)14-s + (−0.241 − 0.970i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.882 + 0.469i)20-s + (−0.882 + 0.469i)22-s + (−0.0348 + 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.0258 + 0.999i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.0258 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04356954632 - 0.04245771291i\)
\(L(\frac12)\) \(\approx\) \(-0.04356954632 - 0.04245771291i\)
\(L(1)\) \(\approx\) \(0.6291383241 - 0.4134464807i\)
\(L(1)\) \(\approx\) \(0.6291383241 - 0.4134464807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (-0.0348 - 0.999i)T \)
13 \( 1 + (0.559 + 0.829i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.0348 + 0.999i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (0.438 + 0.898i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.997 + 0.0697i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.848 - 0.529i)T \)
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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

−22.77609000298884394090593792681, −21.99409543295465350831601578498, −20.935527761187625749076217793319, −20.021731295685825528153175209685, −19.0146318507227637882577740228, −18.3133740137304842391505392425, −17.88267994661019860191111192556, −17.08304068589794944220346547365, −15.97893446612470776215317817445, −15.126560358414455258059092811633, −14.79930286919580007327468360435, −14.05008281768578353062825891277, −12.90324118903718696916854731462, −11.9025213143047930886214440848, −10.667260684125565347039886003, −10.39651350064089338516111933138, −9.14537707103376997377276109901, −8.31679652740177915617731652649, −7.51443555803461387496917134104, −6.831305551992048836429585993459, −5.80635989368994070578242000593, −5.02400260515864656386967788516, −3.95954059555106344526393563598, −2.59149402485156277272807747061, −1.48935904500956698493525159522, 0.01618991593046512482198379081, 1.25573789690524348976361549510, 1.69983409300884601159482378011, 3.35394452239047729178620442837, 4.08336319985504270898400551260, 4.926784070811901277278603925675, 6.02151539612346743378575402038, 7.60747936875946762246037059538, 8.21347166664555874769511726313, 8.85741010384028162486916591321, 9.82247730154530034122248471327, 10.746238011712428919785562740899, 11.60166921641322771712073133465, 12.03234805441131549522612982236, 13.30882945820232408037842813636, 13.6360385878323062441732795308, 14.70595118370566956591825218714, 16.07756701291375393530348845201, 16.83260200487728005355130297069, 17.17686796331668384670792543132, 18.33381657030980222610632238232, 19.071041799940429666905015822832, 19.719038058675059599204965740971, 20.73594070040298608136347531146, 21.14435918131626554769233704685

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