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Properties

Label 1-837-837.211-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.832 + 0.554i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + 17-s + (−0.5 − 0.866i)19-s + (0.766 + 0.642i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + 17-s + (−0.5 − 0.866i)19-s + (0.766 + 0.642i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.832 + 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2795210902 - 0.9233743951i\)
\(L(\frac12)\) \(\approx\) \(-0.2795210902 - 0.9233743951i\)
\(L(1)\) \(\approx\) \(0.5426009565 - 0.7033947860i\)
\(L(1)\) \(\approx\) \(0.5426009565 - 0.7033947860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−22.93790829920366788525905438166, −21.939875640694783630734810082721, −21.30854428164351125543199610951, −20.20405667262241832998589474945, −19.12742451415380360807816527542, −18.44322257744090815977976263208, −17.9867722002471993522661394528, −16.713468150729921582622048784986, −16.12117778895505281773922771506, −15.33858582771681524350132185597, −14.67987132487743595391619165763, −14.13626580218933184470800990099, −12.739837831617862575105246735887, −12.216216612071731858242033306374, −11.41801023932726726889626560454, −10.09106174774072470267657032157, −9.207676615367139353633210380164, −8.23689005484238027986657768347, −7.743687598910512814422693765034, −6.66937581984665671939995375556, −5.96168583482975737174752586078, −4.87510911381457181264032661248, −4.029549035699780651224572515, −3.16631399880366833335798775406, −1.62835854754914343868293388592, 0.48394942690683491190607863022, 1.231716236662423631446878509217, 2.83487595861323451039271446472, 3.72848300904082654645183362569, 4.26653286085631802008983425116, 5.34822994791832354319972787484, 6.43021109922029242548751440238, 7.976304228825323436498191744242, 8.21802116243810890104988520393, 9.409873251142760890163008837919, 10.42465469777515193763704456146, 11.11141096128315099257366901685, 11.69049131684138389031946384185, 12.70947005529559737326655129279, 13.40652209432363653789820155216, 14.16098123049558390037482613040, 15.073645110815418239343942450221, 16.14255050268134029025633882242, 16.86840519319257338713749252069, 17.84818612842030176446563297646, 18.7225239251309307747179809772, 19.5461401484713571070215492999, 19.97680094291757290367642240896, 20.85383270058277845501886021589, 21.40902132411900585842731349750

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