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Properties

Label 2-320-4.3-c10-0-68
Degree 22
Conductor 320320
Sign −1-1
Analytic cond. 203.314203.314
Root an. cond. 14.258814.2588
Motivic weight 1010
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 206. i·3-s − 1.39e3·5-s + 1.52e3i·7-s + 1.64e4·9-s − 2.94e5i·11-s − 1.98e5·13-s + 2.88e5i·15-s + 3.67e5·17-s + 9.19e5i·19-s + 3.15e5·21-s − 2.57e6i·23-s + 1.95e6·25-s − 1.55e7i·27-s + 1.40e7·29-s − 4.90e7i·31-s + ⋯
L(s)  = 1  − 0.849i·3-s − 0.447·5-s + 0.0908i·7-s + 0.278·9-s − 1.82i·11-s − 0.533·13-s + 0.379i·15-s + 0.259·17-s + 0.371i·19-s + 0.0771·21-s − 0.400i·23-s + 0.200·25-s − 1.08i·27-s + 0.686·29-s − 1.71i·31-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=(− Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+5) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: −1-1
Analytic conductor: 203.314203.314
Root analytic conductor: 14.258814.2588
Motivic weight: 1010
Rational: no
Arithmetic: yes
Character: χ320(191,⋅)\chi_{320} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :5), −1)(2,\ 320,\ (\ :5),\ -1)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 1.6501703431.650170343
L(12)L(\frac12) ≈\approx 1.6501703431.650170343
L(6)L(6) not available
L(1)L(1) not available

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
5 1+1.39e3T 1 + 1.39e3T
good3 1+206.iT−5.90e4T2 1 + 206. iT - 5.90e4T^{2}
7 1−1.52e3iT−2.82e8T2 1 - 1.52e3iT - 2.82e8T^{2}
11 1+2.94e5iT−2.59e10T2 1 + 2.94e5iT - 2.59e10T^{2}
13 1+1.98e5T+1.37e11T2 1 + 1.98e5T + 1.37e11T^{2}
17 1−3.67e5T+2.01e12T2 1 - 3.67e5T + 2.01e12T^{2}
19 1−9.19e5iT−6.13e12T2 1 - 9.19e5iT - 6.13e12T^{2}
23 1+2.57e6iT−4.14e13T2 1 + 2.57e6iT - 4.14e13T^{2}
29 1−1.40e7T+4.20e14T2 1 - 1.40e7T + 4.20e14T^{2}
31 1+4.90e7iT−8.19e14T2 1 + 4.90e7iT - 8.19e14T^{2}
37 1−1.21e8T+4.80e15T2 1 - 1.21e8T + 4.80e15T^{2}
41 1+1.48e8T+1.34e16T2 1 + 1.48e8T + 1.34e16T^{2}
43 1−1.41e8iT−2.16e16T2 1 - 1.41e8iT - 2.16e16T^{2}
47 1+3.99e7iT−5.25e16T2 1 + 3.99e7iT - 5.25e16T^{2}
53 1+1.03e8T+1.74e17T2 1 + 1.03e8T + 1.74e17T^{2}
59 1−4.38e8iT−5.11e17T2 1 - 4.38e8iT - 5.11e17T^{2}
61 1−2.83e8T+7.13e17T2 1 - 2.83e8T + 7.13e17T^{2}
67 1+2.00e9iT−1.82e18T2 1 + 2.00e9iT - 1.82e18T^{2}
71 1+1.54e9iT−3.25e18T2 1 + 1.54e9iT - 3.25e18T^{2}
73 1−3.44e9T+4.29e18T2 1 - 3.44e9T + 4.29e18T^{2}
79 1+2.88e9iT−9.46e18T2 1 + 2.88e9iT - 9.46e18T^{2}
83 1+3.41e9iT−1.55e19T2 1 + 3.41e9iT - 1.55e19T^{2}
89 1+2.47e9T+3.11e19T2 1 + 2.47e9T + 3.11e19T^{2}
97 1+2.70e9T+7.37e19T2 1 + 2.70e9T + 7.37e19T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.363567402845948743609862837273, −8.178758067520467644754310812291, −7.75634306706940912707963587432, −6.54535247442954319744243388099, −5.84340815815117005496716708897, −4.48702046495195020177768463533, −3.34788021856252720991722453478, −2.30471428413957611590568213672, −0.987250344852275111573504732314, −0.35864366545543602859634674520, 1.18206625053966884765761118923, 2.41470957735339896138782035271, 3.68926644035856057278198933630, 4.57023093202551840745113908411, 5.14962332969057042488081699888, 6.85486380712241787155762975709, 7.40300554045692841758574268388, 8.638746542663406596977379385377, 9.787336117749519584969648616172, 10.06920885593579486543054002413

Graph of the ZZ-function along the critical line