Properties

Label 1-4160-4160.2629-r0-0-0
Degree 11
Conductor 41604160
Sign 0.5870.809i0.587 - 0.809i
Analytic cond. 19.318919.3189
Root an. cond. 19.318919.3189
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  + (0.991 + 0.130i)3-s + (0.258 − 0.965i)7-s + (0.965 + 0.258i)9-s + (−0.608 − 0.793i)11-s + (0.866 − 0.5i)17-s + (0.793 + 0.608i)19-s + (0.382 − 0.923i)21-s + (−0.258 − 0.965i)23-s + (0.923 + 0.382i)27-s + (0.991 + 0.130i)29-s − 31-s + (−0.5 − 0.866i)33-s + (−0.793 + 0.608i)37-s + (0.258 + 0.965i)41-s + (0.991 − 0.130i)43-s + ⋯
L(s)  = 1  + (0.991 + 0.130i)3-s + (0.258 − 0.965i)7-s + (0.965 + 0.258i)9-s + (−0.608 − 0.793i)11-s + (0.866 − 0.5i)17-s + (0.793 + 0.608i)19-s + (0.382 − 0.923i)21-s + (−0.258 − 0.965i)23-s + (0.923 + 0.382i)27-s + (0.991 + 0.130i)29-s − 31-s + (−0.5 − 0.866i)33-s + (−0.793 + 0.608i)37-s + (0.258 + 0.965i)41-s + (0.991 − 0.130i)43-s + ⋯

Functional equation

Λ(s)=(4160s/2ΓR(s)L(s)=((0.5870.809i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4160s/2ΓR(s)L(s)=((0.5870.809i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 41604160    =    265132^{6} \cdot 5 \cdot 13
Sign: 0.5870.809i0.587 - 0.809i
Analytic conductor: 19.318919.3189
Root analytic conductor: 19.318919.3189
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4160(2629,)\chi_{4160} (2629, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 4160, (0: ), 0.5870.809i)(1,\ 4160,\ (0:\ ),\ 0.587 - 0.809i)

Particular Values

L(12)L(\frac{1}{2}) \approx 2.5403347491.294578236i2.540334749 - 1.294578236i
L(12)L(\frac12) \approx 2.5403347491.294578236i2.540334749 - 1.294578236i
L(1)L(1) \approx 1.5868425020.2765977031i1.586842502 - 0.2765977031i
L(1)L(1) \approx 1.5868425020.2765977031i1.586842502 - 0.2765977031i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
13 1 1
good3 1+(0.991+0.130i)T 1 + (0.991 + 0.130i)T
7 1+(0.2580.965i)T 1 + (0.258 - 0.965i)T
11 1+(0.6080.793i)T 1 + (-0.608 - 0.793i)T
17 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
19 1+(0.793+0.608i)T 1 + (0.793 + 0.608i)T
23 1+(0.2580.965i)T 1 + (-0.258 - 0.965i)T
29 1+(0.991+0.130i)T 1 + (0.991 + 0.130i)T
31 1T 1 - T
37 1+(0.793+0.608i)T 1 + (-0.793 + 0.608i)T
41 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
43 1+(0.9910.130i)T 1 + (0.991 - 0.130i)T
47 1iT 1 - iT
53 1+(0.3820.923i)T 1 + (0.382 - 0.923i)T
59 1+(0.1300.991i)T 1 + (-0.130 - 0.991i)T
61 1+(0.6080.793i)T 1 + (0.608 - 0.793i)T
67 1+(0.991+0.130i)T 1 + (0.991 + 0.130i)T
71 1+(0.258+0.965i)T 1 + (-0.258 + 0.965i)T
73 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
79 1+iT 1 + iT
83 1+(0.9230.382i)T 1 + (0.923 - 0.382i)T
89 1+(0.965+0.258i)T 1 + (-0.965 + 0.258i)T
97 1+(0.50.866i)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

−18.529964529272654646716433315119, −17.93925993016366482759483219997, −17.427032181782684822944420970204, −16.160295203586301733756803639004, −15.67437231199667731029742408088, −15.1106517998476027282280957474, −14.50472341825260143631553237803, −13.797243557377629392564858739467, −13.12802216882438172250565888797, −12.232701897429831745190308492349, −12.07208196386280635052749454501, −10.82904522672209050337295424100, −10.12770084606912539917039245565, −9.3517898780635661841314695904, −8.91761872790421031105749328262, −8.01999418385559123315489369868, −7.52701959550597445408694666719, −6.84634009050970244673327523946, −5.64425956909201189137886232858, −5.21285665524734845772828085975, −4.17042032135467262612176873726, −3.40488872036763717159469516348, −2.56850325664154634364326071128, −2.025466599980187138656698055310, −1.127844251308407303791490559523, 0.74066883754819810151392704620, 1.52104565341009899135081855406, 2.60889451075428999348807251542, 3.276829856714278080352161804089, 3.890002486645476301645578793269, 4.77875024375957074507518074605, 5.46584712016367970423273593591, 6.587958510043321134164045688593, 7.290854628992893051727153972713, 8.108145627309084580669820681233, 8.26960835229514446766097979177, 9.43721884467812678630483880242, 9.97473728821821574857698928849, 10.6294282571066314975628846171, 11.29524739045764382246484877464, 12.35889836074532788838798216189, 12.94577295083900879101760170283, 13.84194159845455492102681426546, 14.14785656911669440937063211134, 14.635915338618845187904925847378, 15.74992602995279451559522347456, 16.19806677019779967573267709250, 16.72778348978788500084932012585, 17.76669593271242013847069615070, 18.44874116265442339003469185957

Graph of the ZZ-function along the critical line