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Properties

Label 1-2e9-512.21-r0-0-0
Degree 11
Conductor 512512
Sign −0.460−0.887i-0.460 - 0.887i
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  + (0.336 − 0.941i)3-s + (0.595 − 0.803i)5-s + (0.995 + 0.0980i)7-s + (−0.773 − 0.634i)9-s + (−0.740 + 0.671i)11-s + (−0.989 − 0.146i)13-s + (−0.555 − 0.831i)15-s + (0.555 − 0.831i)17-s + (−0.857 − 0.514i)19-s + (0.427 − 0.903i)21-s + (0.881 − 0.471i)23-s + (−0.290 − 0.956i)25-s + (−0.857 + 0.514i)27-s + (−0.0490 − 0.998i)29-s + (0.382 − 0.923i)31-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)3-s + (0.595 − 0.803i)5-s + (0.995 + 0.0980i)7-s + (−0.773 − 0.634i)9-s + (−0.740 + 0.671i)11-s + (−0.989 − 0.146i)13-s + (−0.555 − 0.831i)15-s + (0.555 − 0.831i)17-s + (−0.857 − 0.514i)19-s + (0.427 − 0.903i)21-s + (0.881 − 0.471i)23-s + (−0.290 − 0.956i)25-s + (−0.857 + 0.514i)27-s + (−0.0490 − 0.998i)29-s + (0.382 − 0.923i)31-s + ⋯

Functional equation

Λ(s)=(512s/2 ΓR(s) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(512s/2 ΓR(s) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 512512    =    292^{9}
Sign: −0.460−0.887i-0.460 - 0.887i
Analytic conductor: 2.377712.37771
Root analytic conductor: 2.377712.37771
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ512(21,⋅)\chi_{512} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 512, (0: ), −0.460−0.887i)(1,\ 512,\ (0:\ ),\ -0.460 - 0.887i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.8238693918−1.355609992i0.8238693918 - 1.355609992i
L(12)L(\frac12) ≈\approx 0.8238693918−1.355609992i0.8238693918 - 1.355609992i
L(1)L(1) ≈\approx 1.088574002−0.6556388801i1.088574002 - 0.6556388801i
L(1)L(1) ≈\approx 1.088574002−0.6556388801i1.088574002 - 0.6556388801i

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+(0.336−0.941i)T 1 + (0.336 - 0.941i)T
5 1+(0.595−0.803i)T 1 + (0.595 - 0.803i)T
7 1+(0.995+0.0980i)T 1 + (0.995 + 0.0980i)T
11 1+(−0.740+0.671i)T 1 + (-0.740 + 0.671i)T
13 1+(−0.989−0.146i)T 1 + (-0.989 - 0.146i)T
17 1+(0.555−0.831i)T 1 + (0.555 - 0.831i)T
19 1+(−0.857−0.514i)T 1 + (-0.857 - 0.514i)T
23 1+(0.881−0.471i)T 1 + (0.881 - 0.471i)T
29 1+(−0.0490−0.998i)T 1 + (-0.0490 - 0.998i)T
31 1+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
37 1+(−0.242+0.970i)T 1 + (-0.242 + 0.970i)T
41 1+(−0.290+0.956i)T 1 + (-0.290 + 0.956i)T
43 1+(0.941−0.336i)T 1 + (0.941 - 0.336i)T
47 1+(−0.980+0.195i)T 1 + (-0.980 + 0.195i)T
53 1+(−0.0490+0.998i)T 1 + (-0.0490 + 0.998i)T
59 1+(0.989−0.146i)T 1 + (0.989 - 0.146i)T
61 1+(−0.427−0.903i)T 1 + (-0.427 - 0.903i)T
67 1+(0.903−0.427i)T 1 + (0.903 - 0.427i)T
71 1+(−0.634−0.773i)T 1 + (-0.634 - 0.773i)T
73 1+(0.995−0.0980i)T 1 + (0.995 - 0.0980i)T
79 1+(−0.195+0.980i)T 1 + (-0.195 + 0.980i)T
83 1+(−0.242−0.970i)T 1 + (-0.242 - 0.970i)T
89 1+(0.881+0.471i)T 1 + (0.881 + 0.471i)T
97 1+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)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

−23.806205999647170083641235859110, −22.909349772111888410940121860064, −21.87952821185660127067707133619, −21.2448260276748587356546042782, −20.9800615975193691058889741529, −19.55794287396572478578675269463, −18.94267996258833460802801694671, −17.7437680602367887795397797812, −17.13378636036067951458671911161, −16.18142787131515011484880222812, −15.00560076350364412041818029279, −14.574770601268465365494053652743, −13.91746832600247952171587987028, −12.726157760100381486185640043038, −11.35754830545942553141128767507, −10.61537892474107542193447332300, −10.14610375895958674792940057764, −8.93523302221027800427035607192, −8.106899146906628317185087274991, −7.09069142990538479896491068252, −5.6560803107969176721250684539, −5.072525181757765808799392069823, −3.81113170842580601174050332595, −2.82142586096585599710823016987, −1.83724252977227383326552840913, 0.80110034793448695510897490364, 2.055954015165165765157311845705, 2.65003438750313558942060445748, 4.621695657406796554505913498402, 5.18925456506212216204512892234, 6.399541335118512441430020263929, 7.56015990676033222250216480247, 8.13020046101171368930615328844, 9.146208858170559950379046166258, 10.027244003692111737357206171765, 11.37321368879314180147900150207, 12.25298275520320156839836279862, 12.94800219234592017160553853528, 13.727346395391316632721934902036, 14.65153141605122937768493253775, 15.381262493566073607934431066553, 16.97134580449205091624688740670, 17.32746073784830316118500716667, 18.20514818747061255763861062613, 18.98589438444825692524994598899, 20.087789846718307481625061396170, 20.74624884514919969622902376097, 21.29907867004172503850878999297, 22.63369768994572129618209519247, 23.56009421156221476573049835791

Graph of the ZZ-function along the critical line