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Properties

Label 1-837-837.824-r1-0-0
Degree $1$
Conductor $837$
Sign $0.665 + 0.746i$
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  + (−0.0348 − 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (0.241 + 0.970i)11-s + (0.848 + 0.529i)13-s + (0.719 + 0.694i)14-s + (0.990 − 0.139i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.961 − 0.275i)20-s + (0.961 − 0.275i)22-s + (0.241 − 0.970i)23-s + ⋯
L(s)  = 1  + (−0.0348 − 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (0.241 + 0.970i)11-s + (0.848 + 0.529i)13-s + (0.719 + 0.694i)14-s + (0.990 − 0.139i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.961 − 0.275i)20-s + (0.961 − 0.275i)22-s + (0.241 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.665 + 0.746i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.729184318 + 0.7749740569i\)
\(L(\frac12)\) \(\approx\) \(1.729184318 + 0.7749740569i\)
\(L(1)\) \(\approx\) \(1.096624551 - 0.1364760264i\)
\(L(1)\) \(\approx\) \(1.096624551 - 0.1364760264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0348 - 0.999i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (0.848 + 0.529i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.241 - 0.970i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (0.0348 + 0.999i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (0.882 + 0.469i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−22.0781018260640437617992037047, −21.11086438975327316863321551603, −20.33578799806681002990940843388, −19.21615297804532953758358364368, −18.46222264696088699392310064548, −17.65660429305771168781201879859, −16.919425057976561336759577355657, −16.179408166384251448921832442806, −15.77423449964392257336308037878, −14.257853872961824036497901659624, −13.90799945503019828458786377051, −13.20856916211590261103126043760, −12.43324083783229382087388562679, −10.93422374855078407665263363811, −10.08064547269822877939399108537, −9.28512708965545099894901769650, −8.62702399224757436567288806936, −7.52372479611849613070953898666, −6.69042060770803179006665491925, −5.762648516471522167063291911461, −5.33820345349343446528390770832, −3.9017171574719060709171437054, −3.18144690898570607015554600673, −1.32086773884357426466825374629, −0.47857314956264810556471068577, 1.17100938479519227588570880216, 2.15866975100683046800933991162, 2.85096501903983322882350324249, 3.97030062550332223715976826079, 4.98244230598307766040186749197, 6.06579273055890761225978199702, 6.74044830889601756651080419083, 8.280446518496868071756073417562, 9.172246567182114956051369010474, 9.70495511160673526861136115073, 10.50134652825890679228405468326, 11.38759972004196158209554608721, 12.33722246586652172113295963737, 13.03752156813442339984960329160, 13.67350729314560161302464421163, 14.66188456798719011793765499674, 15.41506478801812876858814567469, 16.73805061784120067567217796360, 17.45134793907902017346032303157, 18.29412866858184448290451270836, 18.80143228946449407515566277134, 19.67927543078011774400546571137, 20.51161801569983155583131731326, 21.375441971648277611189362711866, 21.843267174149276456979476050193

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