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Properties

Label 1-2e9-512.325-r0-0-0
Degree 11
Conductor 512512
Sign 0.715−0.698i0.715 - 0.698i
Analytic cond. 2.377712.37771
Root an. cond. 2.377712.37771
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  + (0.803 − 0.595i)3-s + (0.740 − 0.671i)5-s + (−0.471 + 0.881i)7-s + (0.290 − 0.956i)9-s + (0.970 + 0.242i)11-s + (0.998 − 0.0490i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.336 + 0.941i)19-s + (0.146 + 0.989i)21-s + (−0.634 + 0.773i)23-s + (0.0980 − 0.995i)25-s + (−0.336 − 0.941i)27-s + (−0.857 − 0.514i)29-s + (0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)3-s + (0.740 − 0.671i)5-s + (−0.471 + 0.881i)7-s + (0.290 − 0.956i)9-s + (0.970 + 0.242i)11-s + (0.998 − 0.0490i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.336 + 0.941i)19-s + (0.146 + 0.989i)21-s + (−0.634 + 0.773i)23-s + (0.0980 − 0.995i)25-s + (−0.336 − 0.941i)27-s + (−0.857 − 0.514i)29-s + (0.923 + 0.382i)31-s + ⋯

Functional equation

Λ(s)=(512s/2 ΓR(s) L(s)=((0.715−0.698i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(512s/2 ΓR(s) L(s)=((0.715−0.698i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 512512    =    292^{9}
Sign: 0.715−0.698i0.715 - 0.698i
Analytic conductor: 2.377712.37771
Root analytic conductor: 2.377712.37771
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ512(325,⋅)\chi_{512} (325, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 512, (0: ), 0.715−0.698i)(1,\ 512,\ (0:\ ),\ 0.715 - 0.698i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.952031111−0.7945606306i1.952031111 - 0.7945606306i
L(12)L(\frac12) ≈\approx 1.952031111−0.7945606306i1.952031111 - 0.7945606306i
L(1)L(1) ≈\approx 1.520638985−0.3769512766i1.520638985 - 0.3769512766i
L(1)L(1) ≈\approx 1.520638985−0.3769512766i1.520638985 - 0.3769512766i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
good3 1+(0.803−0.595i)T 1 + (0.803 - 0.595i)T
5 1+(0.740−0.671i)T 1 + (0.740 - 0.671i)T
7 1+(−0.471+0.881i)T 1 + (-0.471 + 0.881i)T
11 1+(0.970+0.242i)T 1 + (0.970 + 0.242i)T
13 1+(0.998−0.0490i)T 1 + (0.998 - 0.0490i)T
17 1+(−0.195−0.980i)T 1 + (-0.195 - 0.980i)T
19 1+(−0.336+0.941i)T 1 + (-0.336 + 0.941i)T
23 1+(−0.634+0.773i)T 1 + (-0.634 + 0.773i)T
29 1+(−0.857−0.514i)T 1 + (-0.857 - 0.514i)T
31 1+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
37 1+(0.903+0.427i)T 1 + (0.903 + 0.427i)T
41 1+(0.0980+0.995i)T 1 + (0.0980 + 0.995i)T
43 1+(0.595−0.803i)T 1 + (0.595 - 0.803i)T
47 1+(0.555−0.831i)T 1 + (0.555 - 0.831i)T
53 1+(−0.857+0.514i)T 1 + (-0.857 + 0.514i)T
59 1+(−0.998−0.0490i)T 1 + (-0.998 - 0.0490i)T
61 1+(−0.146+0.989i)T 1 + (-0.146 + 0.989i)T
67 1+(−0.989−0.146i)T 1 + (-0.989 - 0.146i)T
71 1+(−0.956+0.290i)T 1 + (-0.956 + 0.290i)T
73 1+(−0.471−0.881i)T 1 + (-0.471 - 0.881i)T
79 1+(0.831−0.555i)T 1 + (0.831 - 0.555i)T
83 1+(0.903−0.427i)T 1 + (0.903 - 0.427i)T
89 1+(−0.634−0.773i)T 1 + (-0.634 - 0.773i)T
97 1+(0.382−0.923i)T 1 + (0.382 - 0.923i)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

−23.710894488675774570722624609244, −22.49389961290255045198718931810, −22.077105976399361850472812886044, −21.137387421160799684965127336525, −20.36367920987201069115897037914, −19.50000548400737722632400643365, −18.86659551649904433075815610021, −17.64595256129727040522727982587, −16.855871140243039820048954780620, −15.992923382965814505790816337, −14.97554107512792752543384030623, −14.20930918948399662794117643138, −13.59483170808698545154740043871, −12.81575588702527007132782312818, −11.01675081913047942286305037345, −10.6804765573049366947889978822, −9.60220325844854569284511363833, −8.968871882662774147304718406945, −7.84391209418079589821920856909, −6.671283589126808655620955788936, −6.01806834152786528194285766686, −4.35048596379107347157051456292, −3.70072151433654722343733063833, −2.67642742786918101917185417992, −1.46987439019017598702920268189, 1.2520942139772787537531068619, 2.09813837963228003000494541019, 3.22779915796836636575614619514, 4.359018292145722649081301618489, 5.90113863749781679793902566641, 6.31570265571217896612673272417, 7.66314571071652933108852776373, 8.72294787063540716765890027008, 9.231601651061124164617223081008, 9.98930374631863561870948851231, 11.71558224018077010432451206406, 12.29637488442720004873869634633, 13.306397887647039251430384828428, 13.80956536146090803775389120687, 14.82768402637411303572319277015, 15.74798172562540260517517380440, 16.66818344226364484411952245290, 17.764857806720726613750101906493, 18.4309994835945472216012202211, 19.24200158420523177610875836732, 20.19570681781799109767000437597, 20.805540971692946044279078094301, 21.6674760293497858304525119007, 22.62317040208497718858316056868, 23.62707692095969159257825009543

Graph of the ZZ-function along the critical line