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Properties

Label 1-837-837.59-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.634 + 0.773i$
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  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (0.719 − 0.694i)11-s + (0.848 + 0.529i)13-s + (−0.961 + 0.275i)14-s + (−0.374 + 0.927i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.241 + 0.970i)20-s + (0.961 − 0.275i)22-s + (−0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (0.719 − 0.694i)11-s + (0.848 + 0.529i)13-s + (−0.961 + 0.275i)14-s + (−0.374 + 0.927i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.241 + 0.970i)20-s + (0.961 − 0.275i)22-s + (−0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.634 + 0.773i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.914828515 + 4.048495524i\)
\(L(\frac12)\) \(\approx\) \(1.914828515 + 4.048495524i\)
\(L(1)\) \(\approx\) \(1.764813758 + 1.161441749i\)
\(L(1)\) \(\approx\) \(1.764813758 + 1.161441749i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
11 \( 1 + (0.719 - 0.694i)T \)
13 \( 1 + (0.848 + 0.529i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
37 \( 1 + T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (0.0348 + 0.999i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (-0.0348 - 0.999i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−21.72015112640481082137183687018, −20.84234395331597136546451170728, −20.25560512196894993974101405606, −19.55705490512856458220559428507, −18.64737004096776916560125579428, −17.54435142856806256692729244211, −16.82044919785775207898259109657, −15.95592304062737144385626804380, −15.01211103343669887323955639460, −14.096059768250097482890788903151, −13.55781958927925487622087012092, −12.67313450495822890027147504956, −12.254635292629042372320439007672, −10.88609021543401746146738696874, −10.19446225585606239004697403627, −9.65837017225965378771240013041, −8.48873321818854214498537732833, −7.080299748437882898940024821537, −6.224437775909780654695173521179, −5.73057509560601108090921570322, −4.39981799468411557363371575528, −3.823699251711898497801424407553, −2.63215733592572911589816516599, −1.61903767282540059798967134936, −0.68919396817724530034555963065, 1.44006491952067675768355336651, 2.61834035803842044358600798553, 3.33854802411379995665100955621, 4.39408553982655369734253149610, 5.67305486701163628539341523014, 6.18842796990639595062825216908, 6.686947171345997030817966308224, 8.05238936356017720447064302537, 8.95942894715983687556413529966, 9.78866450533702765410247566349, 10.97703575841310079844907150984, 11.77434140838505744511049526058, 12.67932723762589361492195869274, 13.41380813123315844533399785001, 14.23071483307708987550410293145, 14.66954386171588140715999896308, 15.96221058217821191000343541940, 16.33196396491029580275977247432, 17.238126147539077875348926250429, 18.20493636281423424467355694479, 18.96020369677700239977448642636, 19.95730858181889247330637392258, 21.12857355530811935122636369005, 21.537430769710228894781755191495, 22.15922107650051974101593882799

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