Properties

Label 1-333-333.16-r0-0-0
Degree 11
Conductor 333333
Sign 0.1160.993i-0.116 - 0.993i
Analytic cond. 1.546441.54644
Root an. cond. 1.546441.54644
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 + (0.766 − 0.642i)5-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.939 − 0.342i)13-s + 14-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)19-s + (−0.939 + 0.342i)20-s + (0.173 − 0.984i)22-s + 23-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.939 − 0.342i)13-s + 14-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)19-s + (−0.939 + 0.342i)20-s + (0.173 − 0.984i)22-s + 23-s + ⋯

Functional equation

Λ(s)=(333s/2ΓR(s)L(s)=((0.1160.993i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(333s/2ΓR(s)L(s)=((0.1160.993i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 333333    =    32373^{2} \cdot 37
Sign: 0.1160.993i-0.116 - 0.993i
Analytic conductor: 1.546441.54644
Root analytic conductor: 1.546441.54644
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ333(16,)\chi_{333} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 333, (0: ), 0.1160.993i)(1,\ 333,\ (0:\ ),\ -0.116 - 0.993i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.98678664181.109324692i0.9867866418 - 1.109324692i
L(12)L(\frac12) \approx 0.98678664181.109324692i0.9867866418 - 1.109324692i
L(1)L(1) \approx 1.0383931960.6755684971i1.038393196 - 0.6755684971i
L(1)L(1) \approx 1.0383931960.6755684971i1.038393196 - 0.6755684971i

Euler product

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

−25.155686908813516881758937284349, −24.49656314219346037949741276331, −23.50011219620443818119959319714, −22.658979935775596987993633659119, −21.938588406297954218022923903904, −21.1207971783743502750068018374, −19.76360275119755296537225329362, −18.82154697003463634491043631467, −17.74258804958770484253900122936, −17.04565889517109368223544383165, −16.55813424721194738656738864119, −14.96853626174470553390700725786, −14.4790049629310347306676144758, −13.745001614735229101393307770116, −12.799089470975516342227741895807, −11.50992849448290623165480953217, −10.13741553114765054769053871892, −9.563452434401704370192216623641, −8.25467002932208685376391917224, −7.115186725295542807295391876513, −6.58658183095729945882101608230, −5.43948745266473352266167242973, −4.287819894153456810248999028068, −3.2482993072762183627299923344, −1.42654963444816590623548338629, 1.071894934054544037184761387144, 2.24681802382805703626686859716, 3.20630478861928221861678551105, 4.97547700772600019210421816042, 5.17865719451864113993895686030, 6.668912892563185108798627392381, 8.38611544653446383218269346067, 9.31088412060338763950360699340, 9.71074034665526928196639260679, 11.13107612795282973455433267150, 12.098602661923174354284345637813, 12.59794864718415169073296597769, 13.80530463714977649064562199836, 14.45012923159921027221430537764, 15.63174751048524412852009765211, 17.02169549462131228478859934793, 17.71187167219137112356517056845, 18.57432554783162999972839137316, 19.62597733033401481319124202199, 20.340171915939189166506909191079, 21.28825613405900391803241738438, 21.989503655739647546288364842110, 22.555165525269976631826524906, 23.90546496529967394016084244905, 24.83951794370680020471204241240

Graph of the ZZ-function along the critical line