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Properties

Label 1-837-837.272-r0-0-0
Degree 11
Conductor 837837
Sign 0.995+0.0956i0.995 + 0.0956i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.374 − 0.927i)11-s + (0.241 + 0.970i)13-s + (−0.990 + 0.139i)14-s + (0.438 − 0.898i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.615 − 0.788i)20-s + (0.615 + 0.788i)22-s + (−0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (−0.374 − 0.927i)11-s + (0.241 + 0.970i)13-s + (−0.990 + 0.139i)14-s + (0.438 − 0.898i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (0.615 − 0.788i)20-s + (0.615 + 0.788i)22-s + (−0.374 + 0.927i)23-s + ⋯

Functional equation

Λ(s)=(837s/2 ΓR(s) L(s)=((0.995+0.0956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(837s/2 ΓR(s) L(s)=((0.995+0.0956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 837837    =    33⋅313^{3} \cdot 31
Sign: 0.995+0.0956i0.995 + 0.0956i
Analytic conductor: 3.887013.88701
Root analytic conductor: 3.887013.88701
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ837(272,⋅)\chi_{837} (272, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 837, (0: ), 0.995+0.0956i)(1,\ 837,\ (0:\ ),\ 0.995 + 0.0956i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.308977071+0.06272982273i1.308977071 + 0.06272982273i
L(12)L(\frac12) ≈\approx 1.308977071+0.06272982273i1.308977071 + 0.06272982273i
L(1)L(1) ≈\approx 0.9597817256+0.04667932987i0.9597817256 + 0.04667932987i
L(1)L(1) ≈\approx 0.9597817256+0.04667932987i0.9597817256 + 0.04667932987i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
31 1 1
good2 1+(−0.961+0.275i)T 1 + (-0.961 + 0.275i)T
5 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
7 1+(0.990+0.139i)T 1 + (0.990 + 0.139i)T
11 1+(−0.374−0.927i)T 1 + (-0.374 - 0.927i)T
13 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
17 1+(0.669−0.743i)T 1 + (0.669 - 0.743i)T
19 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
23 1+(−0.374+0.927i)T 1 + (-0.374 + 0.927i)T
29 1+(0.961−0.275i)T 1 + (0.961 - 0.275i)T
37 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
41 1+(0.997+0.0697i)T 1 + (0.997 + 0.0697i)T
43 1+(−0.961+0.275i)T 1 + (-0.961 + 0.275i)T
47 1+(0.997−0.0697i)T 1 + (0.997 - 0.0697i)T
53 1+(0.669−0.743i)T 1 + (0.669 - 0.743i)T
59 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
61 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
67 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
71 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
73 1+(−0.669−0.743i)T 1 + (-0.669 - 0.743i)T
79 1+(−0.848−0.529i)T 1 + (-0.848 - 0.529i)T
83 1+(−0.719−0.694i)T 1 + (-0.719 - 0.694i)T
89 1+(−0.978+0.207i)T 1 + (-0.978 + 0.207i)T
97 1+(0.990+0.139i)T 1 + (0.990 + 0.139i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.67933171823147737500917793431, −21.301719791551694911314751499457, −20.43752842528848990413907761143, −19.87875184101492680506153929319, −18.623545342933129585990994483915, −18.12310451654155405796228726662, −17.40147779397279443916292128337, −17.0184754215994238275520489746, −15.715878486788223584647268794874, −14.938372452345259413297269845625, −14.21107932539415463602430040369, −12.90264895895999041405401262048, −12.41100540869465074252813631671, −11.12118811055281862114104144147, −10.468535499796464138834704522050, −10.02908761498626654707016302797, −8.85772804713676935547897161178, −8.12189550704482818964258620921, −7.29197527490627781003217358230, −6.326133763296656304961745335691, −5.37481321408790289971634708362, −4.16118009040429345527805088816, −2.73065535259713192616794885544, −2.078034906042168375495715857818, −1.04942473605270413535647725092, 1.068204817903956836946934003567, 1.85725609466855173252023105067, 2.85088614158494892235445924809, 4.51887298448954183978387089116, 5.59537096326462089058024352408, 6.11416143612085923359945925109, 7.27736403482996896674523007202, 8.2743935140179306592568047365, 8.78831516636145951290958557537, 9.73741386682866635428715537908, 10.48123586915522774203480163335, 11.42239318598306443384814532492, 12.051265090674565491149562025641, 13.516891625180643105737583646096, 14.12589101495956303039566198982, 14.910135228615913844986453375918, 16.07399277497871958950907045673, 16.633817743746120757902475374756, 17.37078611749928338024726784543, 18.18964320405166473105283365497, 18.68896911777200964880981444829, 19.60711500604620614713929847826, 20.73239108861813419885293987425, 21.18835431515370714320865251511, 21.699273582267264771084222110350

Graph of the ZZ-function along the critical line