↑

Properties

Label 1-3e4-81.59-r1-0-0
Degree 11
Conductor 8181
Sign 0.995−0.0968i0.995 - 0.0968i
Analytic cond. 8.704658.70465
Root an. cond. 8.704658.70465
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (0.835 − 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 − 0.116i)20-s + (−0.686 − 0.727i)22-s + (−0.597 + 0.802i)23-s + ⋯
L(s)  = 1  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (0.835 − 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 − 0.116i)20-s + (−0.686 − 0.727i)22-s + (−0.597 + 0.802i)23-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓR(s+1) L(s)=((0.995−0.0968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(81s/2 ΓR(s+1) L(s)=((0.995−0.0968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 8181    =    343^{4}
Sign: 0.995−0.0968i0.995 - 0.0968i
Analytic conductor: 8.704658.70465
Root analytic conductor: 8.704658.70465
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ81(59,⋅)\chi_{81} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 81, (1: ), 0.995−0.0968i)(1,\ 81,\ (1:\ ),\ 0.995 - 0.0968i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.319144983−0.06400410741i1.319144983 - 0.06400410741i
L(12)L(\frac12) ≈\approx 1.319144983−0.06400410741i1.319144983 - 0.06400410741i
L(1)L(1) ≈\approx 0.9544380923−0.2374136663i0.9544380923 - 0.2374136663i
L(1)L(1) ≈\approx 0.9544380923−0.2374136663i0.9544380923 - 0.2374136663i

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
good2 1+(0.0581−0.998i)T 1 + (0.0581 - 0.998i)T
5 1+(−0.973+0.230i)T 1 + (-0.973 + 0.230i)T
7 1+(0.597+0.802i)T 1 + (0.597 + 0.802i)T
11 1+(0.686−0.727i)T 1 + (0.686 - 0.727i)T
13 1+(0.893+0.448i)T 1 + (0.893 + 0.448i)T
17 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
19 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
23 1+(−0.597+0.802i)T 1 + (-0.597 + 0.802i)T
29 1+(0.835+0.549i)T 1 + (0.835 + 0.549i)T
31 1+(0.396+0.918i)T 1 + (0.396 + 0.918i)T
37 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
41 1+(0.0581+0.998i)T 1 + (0.0581 + 0.998i)T
43 1+(−0.286−0.957i)T 1 + (-0.286 - 0.957i)T
47 1+(−0.396+0.918i)T 1 + (-0.396 + 0.918i)T
53 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
59 1+(0.686+0.727i)T 1 + (0.686 + 0.727i)T
61 1+(−0.993+0.116i)T 1 + (-0.993 + 0.116i)T
67 1+(−0.835+0.549i)T 1 + (-0.835 + 0.549i)T
71 1+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
73 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
79 1+(−0.0581+0.998i)T 1 + (-0.0581 + 0.998i)T
83 1+(0.0581−0.998i)T 1 + (0.0581 - 0.998i)T
89 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
97 1+(0.973+0.230i)T 1 + (0.973 + 0.230i)T
show more
show less
   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

−30.757694830242587105301095518446, −30.20016198965782672277304166906, −27.98452329820870590298644766930, −27.577685651213247609628481554788, −26.4534444275305045454078026627, −25.365358501108140093793714190420, −24.22945546424682803117305005862, −23.30066026019023919857706307905, −22.70927955295718166831115809419, −20.98053577687851797659776638049, −19.83336377940792012896691415748, −18.52573520218044133047923691726, −17.316334251832771652436395209136, −16.41247388598860203728242110934, −15.246536043473930778475365182405, −14.34617002477669217570995121341, −13.03226883722862622200547842025, −11.7256011957965632251892732211, −10.150391582934874880509918135761, −8.509112839843962053942580000126, −7.69876900724556852403759875103, −6.47651925252235632222972295379, −4.706849908844874930677044655247, −3.847220132739422287990209419835, −0.7567806208400374489783704841, 1.366477733955942754005466486664, 3.19330514990299328748306998589, 4.31868310749863679989037567318, 5.96901027668593055310559192606, 8.10961565670858111002636243608, 8.91620371384591359013336783182, 10.64497134596835764517819544867, 11.6092690874616166026455620283, 12.33808469240489253974958697323, 13.97984759220641738599748054075, 14.939171205949179704444809824233, 16.41004124507860243654560774771, 17.99570393082350223099580951004, 18.95725488623569863056239213933, 19.66997022643201349901002664936, 21.12505017034725994572243121132, 21.79357807785331076025281675740, 23.164474090260443314369269159020, 23.88709871907000009727868641357, 25.50032449050005935926447942779, 26.963864995462390272882233914203, 27.64977987975326564451881933145, 28.43681281070633870635319087725, 29.91598671227780470617137708450, 30.60862160010000405265784908855

Graph of the ZZ-function along the critical line