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Properties

Label 1-2e9-512.53-r0-0-0
Degree 11
Conductor 512512
Sign 0.460+0.887i0.460 + 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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.671 + 0.740i)3-s + (0.970 + 0.242i)5-s + (0.773 − 0.634i)7-s + (−0.0980 − 0.995i)9-s + (−0.427 + 0.903i)11-s + (−0.514 + 0.857i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.803 − 0.595i)19-s + (−0.0490 + 0.998i)21-s + (−0.956 − 0.290i)23-s + (0.881 + 0.471i)25-s + (0.803 + 0.595i)27-s + (0.336 − 0.941i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.671 + 0.740i)3-s + (0.970 + 0.242i)5-s + (0.773 − 0.634i)7-s + (−0.0980 − 0.995i)9-s + (−0.427 + 0.903i)11-s + (−0.514 + 0.857i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.803 − 0.595i)19-s + (−0.0490 + 0.998i)21-s + (−0.956 − 0.290i)23-s + (0.881 + 0.471i)25-s + (0.803 + 0.595i)27-s + (0.336 − 0.941i)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+0.887i0.460 + 0.887i
Analytic conductor: 2.377712.37771
Root analytic conductor: 2.377712.37771
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ512(53,⋅)\chi_{512} (53, \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 1.169800123+0.7109437973i1.169800123 + 0.7109437973i
L(12)L(\frac12) ≈\approx 1.169800123+0.7109437973i1.169800123 + 0.7109437973i
L(1)L(1) ≈\approx 1.044958669+0.3220602301i1.044958669 + 0.3220602301i
L(1)L(1) ≈\approx 1.044958669+0.3220602301i1.044958669 + 0.3220602301i

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

−23.67753933801928263577344230619, −22.41074143276344449078136504519, −21.91610896044978019596436690523, −21.025876062138671082856536813, −20.14907314278156148901896138521, −18.88684899557718651988077598946, −18.14015783053698222120333579683, −17.795086374848051276983838911509, −16.69487819601179032771610993875, −16.07253419541625476823472561348, −14.61330434260269654881151742812, −13.916942736798720315774690079489, −13.022735579462874948057249680490, −12.21202465797917428759126271686, −11.40351753320080886121897880439, −10.42366135944965152099819250966, −9.4904727192019272235912843034, −8.16985132175867897852749249258, −7.64985730930116011246995987921, −6.165445305591981114297525936775, −5.53004623540397006579907112987, −4.998599531093023522747520453836, −3.00972354616459021026671988077, −1.99585084502382396322124773023, −0.92675586304239397942345352572, 1.29918185016668184763441681231, 2.50615982154721299732061965282, 4.028822782475792947047068127094, 4.84017288812689189082004552874, 5.662327658869981409400894335721, 6.73346416164682696318543699378, 7.66608167112840759840013973358, 9.10771287400397727317235491037, 10.00849888965134188821028327983, 10.41662296042889896906719392041, 11.51413689871643229974565093971, 12.31361412490509882415801394326, 13.55490473169007162445260077876, 14.4378763906023031829986102749, 15.033679242778571565702838647674, 16.310760332185090960261045987050, 16.93925443744474895722501381210, 17.8460957206994635636421488830, 18.13762181656418803301793357847, 19.70838459594434994531721694234, 20.65735731726078983504345337408, 21.31325689790889715701249495634, 21.88149244413785455059575007969, 22.86990333886266964331996008012, 23.61134951614308744490241541304

Graph of the ZZ-function along the critical line