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Properties

Label 1-837-837.320-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: $\chi_{837} (320, \cdot )$
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

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

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