Properties

Label 1-837-837.110-r0-0-0
Degree $1$
Conductor $837$
Sign $0.897 - 0.441i$
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 + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.848 + 0.529i)11-s + (0.997 + 0.0697i)13-s + (0.882 + 0.469i)14-s + (−0.719 − 0.694i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.0348 + 0.999i)20-s + (−0.0348 − 0.999i)22-s + (0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.809 − 0.587i)10-s + (0.848 + 0.529i)11-s + (0.997 + 0.0697i)13-s + (0.882 + 0.469i)14-s + (−0.719 − 0.694i)16-s + (−0.978 + 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.0348 + 0.999i)20-s + (−0.0348 − 0.999i)22-s + (0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.897 - 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173544107 - 0.2729680967i\)
\(L(\frac12)\) \(\approx\) \(1.173544107 - 0.2729680967i\)
\(L(1)\) \(\approx\) \(0.8942542415 - 0.2389009063i\)
\(L(1)\) \(\approx\) \(0.8942542415 - 0.2389009063i\)

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 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.882 + 0.469i)T \)
11 \( 1 + (0.848 + 0.529i)T \)
13 \( 1 + (0.997 + 0.0697i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.559 - 0.829i)T \)
47 \( 1 + (0.241 - 0.970i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.997 + 0.0697i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.374 + 0.927i)T \)
83 \( 1 + (0.438 - 0.898i)T \)
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

−22.48167578571594581247832847590, −21.49528090476498557275443087790, −20.52844012247833824810114444357, −19.42595772094405988423431195610, −19.03247747797717710097937341585, −18.01348271633528305680841234464, −17.34343470133912694633526375198, −16.6725303159307243770605663407, −15.92447248842482067855041069672, −15.04627877371572617873223353930, −14.05054427047299893584229170289, −13.5819611298302144628479760629, −12.79437178131401952590936247930, −11.102778673583928076669557149205, −10.65103395347174323229888011893, −9.52091882797321341247143394999, −9.15202049748670126052991855352, −8.12032429508768407157764038003, −6.87936926126790216588865236883, −6.352997445074598374935333509, −5.81348422944634590176997330029, −4.45179570507762755876908881627, −3.38969719111285118911246820298, −1.982584439541926058818749380003, −0.84483279778885234176594363345, 1.02952794941673810201264697259, 2.043634585472978031307957024931, 2.88858072532672214658493322866, 4.022791816420298339482265914934, 4.96871516059816808584746796240, 6.40733062450665566863426910808, 6.79829889708367161185108007141, 8.612827082987666771231145633499, 8.85218878040257268383802713442, 9.6938166225690849524927050800, 10.49365796485200133207871169504, 11.36254443001058246722613099460, 12.38632155337505484126712709473, 13.06165927960937215312051017206, 13.54425306277737949369225735145, 14.78936871209040911527187654856, 15.88279720512978578973343148154, 16.76406645936187639689112954923, 17.35695984955343421576496447547, 18.19499127141194924332196699158, 18.85680345744207886960044746418, 19.84374255299818788052398131743, 20.309605285103840112018216303630, 21.34076281034005158149279243087, 21.88683390181896791396919270177

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