Properties

Label 1-837-837.241-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.0956 + 0.995i$
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  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.374 + 0.927i)11-s + (0.241 + 0.970i)13-s + (0.990 − 0.139i)14-s + (0.438 − 0.898i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.615 + 0.788i)20-s + (0.615 + 0.788i)22-s + (0.374 − 0.927i)23-s + ⋯
L(s)  = 1  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.374 + 0.927i)11-s + (0.241 + 0.970i)13-s + (0.990 − 0.139i)14-s + (0.438 − 0.898i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.615 + 0.788i)20-s + (0.615 + 0.788i)22-s + (0.374 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.0956 + 0.995i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.0956 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830443631 + 2.014714295i\)
\(L(\frac12)\) \(\approx\) \(1.830443631 + 2.014714295i\)
\(L(1)\) \(\approx\) \(1.671857792 + 0.1725804171i\)
\(L(1)\) \(\approx\) \(1.671857792 + 0.1725804171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.961 - 0.275i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.374 - 0.927i)T \)
29 \( 1 + (-0.961 + 0.275i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.997 - 0.0697i)T \)
43 \( 1 + (-0.961 + 0.275i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.848 - 0.529i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−21.72247642873719356695866450779, −21.07146496291568151215304785155, −20.15723046180034586627640349355, −19.74856240021290389544496993990, −18.57485907970807855357052791484, −17.464723419385546428297497636158, −16.7898919120108181878481048655, −15.86322620150880671843182814292, −15.22576941372237654145781035411, −14.59403040190476036182630772394, −13.49806489718227168546576308822, −13.01472539502710656326901456319, −11.82139364978378593140809156852, −11.309032619069332497038118319922, −10.76236732710009270228778049324, −8.999344250150024177797527667541, −8.182997545119973765284767364, −7.561897109103773881048621008037, −6.58622364248297704826904519269, −5.42304063176687675362876411940, −4.78413778926315318839559017924, −3.8431631896158839799434554007, −3.074657075198399812795126452764, −1.7176652757111538319884172775, −0.38987054103633722361188388733, 1.48389076913205661323608831801, 2.17860265635840668165894199941, 3.51621878292273308910918855896, 4.39567825923418843058066056962, 4.773243201752678298423953927309, 6.278080998829326915286974823647, 6.88813779001332083903422759456, 7.88358365314610860783899509888, 8.79486742450198971395050436686, 10.16312472767081635765845994166, 11.02286654320546255318054916309, 11.57837085326345308541999732089, 12.31348434942278114166081779756, 13.120258845188766648508350004523, 14.32579543998734331225786819799, 14.84628012802708549393569945490, 15.274777656431290337503199893708, 16.412017394607274760905393702683, 17.18244978013435405486139424918, 18.509329972898180950824235240, 18.99963560971374240458645499351, 20.08332505313070105658048081766, 20.49784896774928593725424582775, 21.478131833323286677165620120521, 22.133519772844306273059663855861

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