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Properties

Label 2-1323-1.1-c3-0-80
Degree 22
Conductor 13231323
Sign −1-1
Analytic cond. 78.059578.0595
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.541·2-s − 7.70·4-s − 16.7·5-s − 8.50·8-s − 9.06·10-s − 26.1·11-s + 34.4·13-s + 57.0·16-s − 81.5·17-s + 96.8·19-s + 128.·20-s − 14.1·22-s + 183.·23-s + 154.·25-s + 18.6·26-s − 42.2·29-s + 279.·31-s + 98.9·32-s − 44.1·34-s − 48.6·37-s + 52.4·38-s + 142.·40-s − 4.73·41-s − 419.·43-s + 201.·44-s + 99.6·46-s + 39.9·47-s + ⋯
L(s)  = 1  + 0.191·2-s − 0.963·4-s − 1.49·5-s − 0.375·8-s − 0.286·10-s − 0.717·11-s + 0.735·13-s + 0.891·16-s − 1.16·17-s + 1.16·19-s + 1.44·20-s − 0.137·22-s + 1.66·23-s + 1.23·25-s + 0.140·26-s − 0.270·29-s + 1.62·31-s + 0.546·32-s − 0.222·34-s − 0.216·37-s + 0.224·38-s + 0.562·40-s − 0.0180·41-s − 1.48·43-s + 0.691·44-s + 0.319·46-s + 0.124·47-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 78.059578.0595
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1323, ( :3/2), −1)(2,\ 1323,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) 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)
bad3 1 1
7 1 1
good2 1−0.541T+8T2 1 - 0.541T + 8T^{2}
5 1+16.7T+125T2 1 + 16.7T + 125T^{2}
11 1+26.1T+1.33e3T2 1 + 26.1T + 1.33e3T^{2}
13 1−34.4T+2.19e3T2 1 - 34.4T + 2.19e3T^{2}
17 1+81.5T+4.91e3T2 1 + 81.5T + 4.91e3T^{2}
19 1−96.8T+6.85e3T2 1 - 96.8T + 6.85e3T^{2}
23 1−183.T+1.21e4T2 1 - 183.T + 1.21e4T^{2}
29 1+42.2T+2.43e4T2 1 + 42.2T + 2.43e4T^{2}
31 1−279.T+2.97e4T2 1 - 279.T + 2.97e4T^{2}
37 1+48.6T+5.06e4T2 1 + 48.6T + 5.06e4T^{2}
41 1+4.73T+6.89e4T2 1 + 4.73T + 6.89e4T^{2}
43 1+419.T+7.95e4T2 1 + 419.T + 7.95e4T^{2}
47 1−39.9T+1.03e5T2 1 - 39.9T + 1.03e5T^{2}
53 1+287.T+1.48e5T2 1 + 287.T + 1.48e5T^{2}
59 1+465.T+2.05e5T2 1 + 465.T + 2.05e5T^{2}
61 1−242.T+2.26e5T2 1 - 242.T + 2.26e5T^{2}
67 1+634.T+3.00e5T2 1 + 634.T + 3.00e5T^{2}
71 1−1.02e3T+3.57e5T2 1 - 1.02e3T + 3.57e5T^{2}
73 1−403.T+3.89e5T2 1 - 403.T + 3.89e5T^{2}
79 1−308.T+4.93e5T2 1 - 308.T + 4.93e5T^{2}
83 1+1.41e3T+5.71e5T2 1 + 1.41e3T + 5.71e5T^{2}
89 1−1.20e3T+7.04e5T2 1 - 1.20e3T + 7.04e5T^{2}
97 1−798.T+9.12e5T2 1 - 798.T + 9.12e5T^{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

−8.690488841863241462368781074409, −8.158273176903584331214449963834, −7.38733927941828277215578655639, −6.41894181264557405710867192107, −5.11144951940966431204644165129, −4.63627158130248088900584063579, −3.64634358677409485460508320678, −2.97367764596240666776793307015, −0.993534174044965387551935864291, 0, 0.993534174044965387551935864291, 2.97367764596240666776793307015, 3.64634358677409485460508320678, 4.63627158130248088900584063579, 5.11144951940966431204644165129, 6.41894181264557405710867192107, 7.38733927941828277215578655639, 8.158273176903584331214449963834, 8.690488841863241462368781074409

Graph of the ZZ-function along the critical line