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Properties

Label 1-837-837.803-r1-0-0
Degree $1$
Conductor $837$
Sign $0.458 + 0.888i$
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  + (0.374 + 0.927i)2-s + (−0.719 + 0.694i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.438 + 0.898i)11-s + (0.990 − 0.139i)13-s + (−0.559 + 0.829i)14-s + (0.0348 − 0.999i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (0.997 − 0.0697i)20-s + (−0.997 − 0.0697i)22-s + (−0.438 − 0.898i)23-s + ⋯
L(s)  = 1  + (0.374 + 0.927i)2-s + (−0.719 + 0.694i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.438 + 0.898i)11-s + (0.990 − 0.139i)13-s + (−0.559 + 0.829i)14-s + (0.0348 − 0.999i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (0.997 − 0.0697i)20-s + (−0.997 − 0.0697i)22-s + (−0.438 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.508975280 + 0.9194127336i\)
\(L(\frac12)\) \(\approx\) \(1.508975280 + 0.9194127336i\)
\(L(1)\) \(\approx\) \(0.9478039852 + 0.4912985002i\)
\(L(1)\) \(\approx\) \(0.9478039852 + 0.4912985002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.374 + 0.927i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (-0.438 + 0.898i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.438 - 0.898i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (-0.374 - 0.927i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (0.615 - 0.788i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.559 + 0.829i)T \)
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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

−21.72377233134072741841805481136, −20.96252087672335044916852166731, −20.27079548149615654227395425463, −19.470118363231598095550480950515, −18.77533375887074938652553781174, −18.10637752140044399989859752456, −17.20395929223858051614667875059, −15.91602153511900657390431739101, −15.33141602308277452762565425922, −14.06306550295304963283906852881, −13.896684246424255979366114658, −12.86104964328984009412458100000, −11.75032501561321994866191385948, −11.09188095785249900786081605846, −10.712131993076729062749455226544, −9.71965994441999891073575742024, −8.39286795384452870544375111719, −7.910647942750537074488232476247, −6.5465919215020549256674392146, −5.68780396451918860307577855043, −4.38577045746660232092954413431, −3.80320945195395056045654943182, −3.00429230133872258212088336371, −1.70974264875369743838056312309, −0.66067901102956825720982882597, 0.56228797740248264842669601373, 2.201349617974075459265521515862, 3.45370636472303800026901352282, 4.63420374880123710698038409614, 4.94164431124077881670602307618, 6.06406799109140026262014740073, 7.106767493008907576671488660256, 7.88662579294362682217535826190, 8.79153492530288808876743689503, 9.12133786527104182901030477432, 10.735531651484267775867825311375, 11.73678827676930219837391967854, 12.46737424316350641280732608847, 13.12507877923746270604192506936, 14.11408875547382926894833489326, 15.08843259416388865678446745699, 15.70277513474917563536143453225, 16.065639900617239949371253957559, 17.24709243270753221043463104584, 18.068148855984513825338797170389, 18.48251625513840082488804093651, 19.8512030464121248113615358146, 20.59241213851862224807754991778, 21.35230945969839773062168509744, 22.30808442507973104452525021573

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