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Properties

Label 1-847-847.102-r0-0-0
Degree 11
Conductor 847847
Sign 0.328+0.944i0.328 + 0.944i
Analytic cond. 3.933453.93345
Root an. cond. 3.933453.93345
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  + (−0.851 − 0.524i)2-s + (0.913 + 0.406i)3-s + (0.449 + 0.893i)4-s + (0.483 + 0.875i)5-s + (−0.564 − 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (0.0475 − 0.998i)10-s + (0.0475 + 0.998i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (−0.969 + 0.244i)19-s + (−0.564 + 0.825i)20-s + ⋯
L(s)  = 1  + (−0.851 − 0.524i)2-s + (0.913 + 0.406i)3-s + (0.449 + 0.893i)4-s + (0.483 + 0.875i)5-s + (−0.564 − 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (0.0475 − 0.998i)10-s + (0.0475 + 0.998i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (−0.969 + 0.244i)19-s + (−0.564 + 0.825i)20-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓR(s) L(s)=((0.328+0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(847s/2 ΓR(s) L(s)=((0.328+0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: 0.328+0.944i0.328 + 0.944i
Analytic conductor: 3.933453.93345
Root analytic conductor: 3.933453.93345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ847(102,⋅)\chi_{847} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 847, (0: ), 0.328+0.944i)(1,\ 847,\ (0:\ ),\ 0.328 + 0.944i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.146432296+0.8147062550i1.146432296 + 0.8147062550i
L(12)L(\frac12) ≈\approx 1.146432296+0.8147062550i1.146432296 + 0.8147062550i
L(1)L(1) ≈\approx 1.034388148+0.2313611660i1.034388148 + 0.2313611660i
L(1)L(1) ≈\approx 1.034388148+0.2313611660i1.034388148 + 0.2313611660i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
11 1 1
good2 1+(−0.851−0.524i)T 1 + (-0.851 - 0.524i)T
3 1+(0.913+0.406i)T 1 + (0.913 + 0.406i)T
5 1+(0.483+0.875i)T 1 + (0.483 + 0.875i)T
13 1+(−0.254−0.967i)T 1 + (-0.254 - 0.967i)T
17 1+(0.640+0.768i)T 1 + (0.640 + 0.768i)T
19 1+(−0.969+0.244i)T 1 + (-0.969 + 0.244i)T
23 1+(0.580+0.814i)T 1 + (0.580 + 0.814i)T
29 1+(−0.466+0.884i)T 1 + (-0.466 + 0.884i)T
31 1+(−0.935−0.353i)T 1 + (-0.935 - 0.353i)T
37 1+(0.988+0.151i)T 1 + (0.988 + 0.151i)T
41 1+(0.610−0.791i)T 1 + (0.610 - 0.791i)T
43 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
47 1+(−0.179+0.983i)T 1 + (-0.179 + 0.983i)T
53 1+(−0.595−0.803i)T 1 + (-0.595 - 0.803i)T
59 1+(0.380−0.924i)T 1 + (0.380 - 0.924i)T
61 1+(0.879+0.475i)T 1 + (0.879 + 0.475i)T
67 1+(0.723+0.690i)T 1 + (0.723 + 0.690i)T
71 1+(0.897−0.441i)T 1 + (0.897 - 0.441i)T
73 1+(0.861+0.508i)T 1 + (0.861 + 0.508i)T
79 1+(0.123−0.992i)T 1 + (0.123 - 0.992i)T
83 1+(−0.736+0.676i)T 1 + (-0.736 + 0.676i)T
89 1+(−0.786−0.618i)T 1 + (-0.786 - 0.618i)T
97 1+(0.516+0.856i)T 1 + (0.516 + 0.856i)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

−21.57102861807976203465531566481, −20.92926679116217684849775058294, −20.18786291186045043650770690735, −19.5160681479416275974126436630, −18.67462173684180124880436864700, −18.14263412422489992544869918766, −16.95320772548779876540025489895, −16.61956878172481077492606927249, −15.573465789371760414829994193156, −14.674282586424450864453537912121, −14.06508484429390729941723635157, −13.169213817650933458528747843761, −12.29535638150461573830771616149, −11.243965593516015509316464796487, −9.963855970018970030265382782249, −9.37227117888218451020526348936, −8.7442857592042625111226217792, −7.99630642948219589097833446202, −7.05636817619625936914479892970, −6.3083646261598136705789614028, −5.16251453698499483225381565078, −4.172958950204818492915511571367, −2.54284837032369662457122532370, −1.84088650071945608332616260458, −0.75051995551404658526664952707, 1.50013315774420348029533978616, 2.38712120849452165773719946220, 3.234012259143567857061608195509, 3.88475675111187864074511742368, 5.43603678883238852798637213804, 6.67121682892140839395235084668, 7.6300617295948618048922809428, 8.19888758856529170507903656177, 9.29760216517226931340407490902, 9.88149203115666877584770238633, 10.6552937906794124819173715971, 11.19495639479496203665000382373, 12.767183601712785514548148583932, 13.080528933534611460787119478511, 14.46480362296007217036480750863, 14.917248525040608912402384406110, 15.83565733395032801576432279038, 16.8541691544441763186154705564, 17.58745443189778268365487064270, 18.49082929494988956749225718194, 19.12726823302753649063728627670, 19.767806341713894377461310540992, 20.6234966107134950993356518254, 21.374461211962857708557494529889, 21.878229715298383859875163239800

Graph of the ZZ-function along the critical line