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Properties

Label 1-837-837.554-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.202 - 0.979i$
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  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.882 − 0.469i)11-s + (−0.559 − 0.829i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.848 + 0.529i)20-s + (0.882 − 0.469i)22-s + (0.848 + 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.882 − 0.469i)11-s + (−0.559 − 0.829i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.848 + 0.529i)20-s + (0.882 − 0.469i)22-s + (0.848 + 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (554, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.202 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5055969984 - 0.6205368767i\)
\(L(\frac12)\) \(\approx\) \(0.5055969984 - 0.6205368767i\)
\(L(1)\) \(\approx\) \(0.7415436014 - 0.1107640349i\)
\(L(1)\) \(\approx\) \(0.7415436014 - 0.1107640349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (-0.882 - 0.469i)T \)
13 \( 1 + (-0.559 - 0.829i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.848 + 0.529i)T \)
29 \( 1 + (0.559 - 0.829i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (-0.438 - 0.898i)T \)
47 \( 1 + (0.241 + 0.970i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.615 + 0.788i)T \)
83 \( 1 + (0.559 - 0.829i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−22.090712544542327748617332229863, −21.37270613108084900147834706699, −20.96406452617766277718055222693, −19.87343797965673259427670345599, −18.98320872579794637794308557610, −18.51010018432769497982036841782, −17.88454324080331823390045468210, −16.99333201033314131357821878424, −16.09288088166265284008889404079, −14.8101834867100263368141835777, −14.51266471116295438059284817677, −13.28574023409924903092011220913, −12.241697368648235065429968484670, −11.783159822952035180854609863858, −10.67900249289408588005297321246, −10.354574954857480249030760996155, −9.27443996373291549606488583276, −8.238504594658110424470541505311, −7.641727780254015495298647156880, −6.704873632596423385046428548876, −5.31131386101141311059521946273, −4.34379507504788826232690425145, −3.260256157307450190662228949286, −2.346067909590792295991330696893, −1.60240448962426809927390670261, 0.468467984848042998794473981290, 1.34176007460556206943214142492, 2.86826204791380871023897242325, 4.534275306093685089940406712965, 5.02798428886313482801134793242, 5.76423427068710533218489519436, 7.20236982043777013081754860726, 7.81076887078763846566374261379, 8.43753265977498978900461750163, 9.36830600359499599553499809931, 10.26133769685238831460945976532, 11.09651440871124643180155459137, 12.09762844153232116446877636211, 13.4405991212354613373407120288, 13.63382033482700919537897899958, 15.011819254319473600971414676206, 15.47476518227971812700416731064, 16.42962886083756947766213509274, 17.120094822048866080747919910317, 17.66994897444876082944597044729, 18.57189767665143690815783466861, 19.48371500118743009244586524794, 20.286419419589821705224340072808, 20.88891642597012946437659210328, 21.92144025022934854383523510835

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