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Properties

Label 1-837-837.592-r1-0-0
Degree $1$
Conductor $837$
Sign $0.446 - 0.894i$
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  + (−0.615 + 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.939 + 0.342i)5-s + (0.438 − 0.898i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.997 − 0.0697i)11-s + (0.374 − 0.927i)13-s + (0.438 + 0.898i)14-s + (−0.882 + 0.469i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (0.559 + 0.829i)20-s + (−0.559 + 0.829i)22-s + (0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (−0.615 + 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.939 + 0.342i)5-s + (0.438 − 0.898i)7-s + (0.913 + 0.406i)8-s + (0.309 − 0.951i)10-s + (0.997 − 0.0697i)11-s + (0.374 − 0.927i)13-s + (0.438 + 0.898i)14-s + (−0.882 + 0.469i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (0.559 + 0.829i)20-s + (−0.559 + 0.829i)22-s + (0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.446 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060750143 - 0.6564315021i\)
\(L(\frac12)\) \(\approx\) \(1.060750143 - 0.6564315021i\)
\(L(1)\) \(\approx\) \(0.7788018650 + 0.03012304063i\)
\(L(1)\) \(\approx\) \(0.7788018650 + 0.03012304063i\)

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 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.438 - 0.898i)T \)
11 \( 1 + (0.997 - 0.0697i)T \)
13 \( 1 + (0.374 - 0.927i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.997 + 0.0697i)T \)
29 \( 1 + (0.615 - 0.788i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.848 - 0.529i)T \)
43 \( 1 + (0.615 - 0.788i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.241 - 0.970i)T \)
83 \( 1 + (-0.990 - 0.139i)T \)
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

−21.85810146301910294056421559554, −21.22099162274473796157736871283, −20.36783008603172927267485850528, −19.60827323538448511803110420455, −19.00660920153479498638662844409, −18.3049604768927267567219054732, −17.39454420984315508826513188857, −16.53611426814832786435878583263, −15.87852498423444469615278211178, −14.84434225728422944503921270917, −13.97078036938440757935584229014, −12.60846406937100218535608692155, −12.28317104962813336043767750914, −11.26490947404472147519556588343, −11.02646373069417561166950604651, −9.43071958313570915066505180023, −8.941102614880691098985551177679, −8.26317048609023183362226170619, −7.30397205975143477414163139865, −6.257453327147522888871391377, −4.73648367340959833689501592421, −4.09560899443274561501202116895, −3.11278522980816732308026639145, −1.87438044788279091689677526110, −1.0585582151332276165859272105, 0.46151496642876262661845413901, 1.117504435391480315600363321436, 2.824454755284324930399206741163, 4.144240645168563757163694672100, 4.73246142247479786022668417711, 6.06506007494647725290678587906, 7.03682200154689702640861969613, 7.4565368225425842501632697979, 8.45383408676455025993498408864, 9.1511450894950970568020451606, 10.34122750111797125256132179616, 11.047227458933951565803284292635, 11.640612107616098832262464260888, 13.12548062138162713836371836528, 13.92295064067898975065202716122, 14.73623969531446979214855816812, 15.497172089289860036203191454174, 16.04998638191714353375560496568, 17.239308034737696271633593497937, 17.51493827145252317365614563735, 18.547395048596618424161612374951, 19.429866443629494505915723954705, 19.96496123313331548767147316012, 20.634190309934830360694110082554, 22.2197536210430814139030688372

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