Properties

Label 1-837-837.211-r0-0-0
Degree 11
Conductor 837837
Sign 0.832+0.554i-0.832 + 0.554i
Analytic cond. 3.887013.88701
Root an. cond. 3.887013.88701
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(837s/2ΓR(s)L(s)=((0.832+0.554i)Λ(1s)\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}
Λ(s)=(837s/2ΓR(s)L(s)=((0.832+0.554i)Λ(1s)\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: 11
Conductor: 837837    =    33313^{3} \cdot 31
Sign: 0.832+0.554i-0.832 + 0.554i
Analytic conductor: 3.887013.88701
Root analytic conductor: 3.887013.88701
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ837(211,)\chi_{837} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 837, (0: ), 0.832+0.554i)(1,\ 837,\ (0:\ ),\ -0.832 + 0.554i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.27952109020.9233743951i-0.2795210902 - 0.9233743951i
L(12)L(\frac12) \approx 0.27952109020.9233743951i-0.2795210902 - 0.9233743951i
L(1)L(1) \approx 0.54260095650.7033947860i0.5426009565 - 0.7033947860i
L(1)L(1) \approx 0.54260095650.7033947860i0.5426009565 - 0.7033947860i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
31 1 1
good2 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
5 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
7 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
11 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
13 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
17 1+T 1 + T
19 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
23 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
29 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
37 1+T 1 + T
41 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
43 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
47 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
53 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
59 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
61 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
67 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
71 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
73 1+T 1 + T
79 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
83 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
89 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
97 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line