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Properties

Label 1-837-837.779-r1-0-0
Degree $1$
Conductor $837$
Sign $0.517 - 0.855i$
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  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.0348 − 0.999i)11-s + (−0.997 + 0.0697i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (0.0348 − 0.999i)22-s + (0.882 − 0.469i)23-s + ⋯
L(s)  = 1  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.0348 − 0.999i)11-s + (−0.997 + 0.0697i)13-s + (−0.848 − 0.529i)14-s + (0.961 + 0.275i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (0.0348 − 0.999i)22-s + (0.882 − 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.517 - 0.855i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.517 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.688553687 - 2.079612341i\)
\(L(\frac12)\) \(\approx\) \(3.688553687 - 2.079612341i\)
\(L(1)\) \(\approx\) \(2.079834475 - 0.2421460496i\)
\(L(1)\) \(\approx\) \(2.079834475 - 0.2421460496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.882 - 0.469i)T \)
11 \( 1 + (-0.0348 - 0.999i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.882 - 0.469i)T \)
29 \( 1 + (0.997 + 0.0697i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (0.559 - 0.829i)T \)
47 \( 1 + (0.719 - 0.694i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (0.997 + 0.0697i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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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

−22.198723187407675148919403952743, −21.359780438842606949599947548201, −20.659194167031028238209312327631, −19.89114827586450340691634924469, −19.12314275193048098223982655582, −17.985877803910789752451644305032, −17.11524404649820864831775785893, −16.349929622108958400938093765847, −15.47843021470447792233820443109, −14.753813513614773465858711965976, −13.87220623583364601045780699421, −13.08597757816929933673740042080, −12.4534955870239908745028192640, −11.86295309865878547460252761066, −10.49504872154977836116727094341, −9.81892792272879511695764039427, −9.11247409814042512582381240186, −7.614842573363137361024906978287, −6.724624271273902299904298495407, −6.01397020719819626998789909622, −5.059660249862620589868389894477, −4.437203398438468433480230729121, −2.94639149858833105612297413065, −2.4132827661342005533040883458, −1.27925958061397872424812315651, 0.60431257831012794263101796334, 2.16033063294458046397054421927, 2.85960339051825814944593346442, 3.79490135310947476873538750609, 4.9307934223628755413334631154, 5.77275614724826889114072684923, 6.70871217168406164762386133590, 7.0471900126499516660547415753, 8.55025820128959998367181101521, 9.57331510319561957937444621229, 10.618614935762891641216483462848, 10.99119700057561306809715534695, 12.38274421924664205191340156738, 12.97777223245022463313274288208, 13.763913005508510892168661506, 14.26587334394451927607796489897, 15.29703353732452287057281725050, 16.04241164712055321452006963713, 17.07008694688553587905393043401, 17.389138977272890814056314462259, 18.93651072711427922790773984487, 19.487629756687927897522567636693, 20.332209725038415426572220359738, 21.38996424720471919042842860473, 21.855939936348992144593429693220

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