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Properties

Label 1-837-837.715-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.999 - 0.0433i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.615 − 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.173 + 0.984i)5-s + (−0.997 − 0.0697i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (0.438 + 0.898i)11-s + (−0.615 + 0.788i)13-s + (0.559 + 0.829i)14-s + (−0.882 − 0.469i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.997 − 0.0697i)20-s + (0.438 − 0.898i)22-s + (−0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (−0.615 − 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.173 + 0.984i)5-s + (−0.997 − 0.0697i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (0.438 + 0.898i)11-s + (−0.615 + 0.788i)13-s + (0.559 + 0.829i)14-s + (−0.882 − 0.469i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.997 − 0.0697i)20-s + (0.438 − 0.898i)22-s + (−0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.999 - 0.0433i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.999 - 0.0433i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002149267152 + 0.09912325492i\)
\(L(\frac12)\) \(\approx\) \(0.002149267152 + 0.09912325492i\)
\(L(1)\) \(\approx\) \(0.5512557060 + 0.006408941986i\)
\(L(1)\) \(\approx\) \(0.5512557060 + 0.006408941986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.615 - 0.788i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
11 \( 1 + (0.438 + 0.898i)T \)
13 \( 1 + (-0.615 + 0.788i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (0.990 - 0.139i)T \)
47 \( 1 + (0.848 - 0.529i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.990 + 0.139i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (-0.615 - 0.788i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−22.02848668172529673865761604673, −20.68436880426527148612789133685, −19.94150444042142454148099007446, −19.3199126920475815557985526177, −18.51846306346126055232290494375, −17.5184881749431799469762422792, −16.82922700764250676421585442493, −16.20352093158815845121251686149, −15.651931570343542213764023084437, −14.55357586299177222526334763471, −13.69075743915333565578522338032, −12.93115379989537481925332602322, −12.05461250922183930667101174292, −10.82671745759595570925561805836, −9.867071208415017598524620494673, −9.26360111807277203095022803253, −8.50524478821704606830681905223, −7.67757601093843655596546843957, −6.56082320115568375487982623308, −5.85327885020251846600656602470, −5.05403377629247561489638265205, −3.94980300468822718974308260989, −2.561234118806957877482060510577, −1.15017331864253056588113257898, −0.05848197325559463353526298396, 1.95916631330477322173917189391, 2.38192440050590045104393181592, 3.75218057865968363638669981778, 4.17575112564332487803925500297, 5.97595226833309315417900652551, 6.88929199645465093533721578297, 7.45735471703523044646956520869, 8.75471171667060159244469657487, 9.606625440277847111357303676379, 10.17515900773157530168080452780, 10.90206613503407193419001362178, 11.95860699195116351770458644036, 12.56660924243454542427475237661, 13.484160824326412078441151112983, 14.417861764861473562021367655, 15.31178152588499932311315932209, 16.32076070429382119086993399350, 17.3198158960734462826676672469, 17.649712670051257683444192409589, 18.95990021776518359304224716351, 19.183208299014394584516948169490, 19.89659732881566565857459920624, 20.94879604481942451142238822554, 21.82286475826558304336721294907, 22.321205025628902834265529575706

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