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Properties

Label 2-20e2-1.1-c5-0-34
Degree 22
Conductor 400400
Sign −1-1
Analytic cond. 64.153564.1535
Root an. cond. 8.009588.00958
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.52·3-s − 68.9·7-s − 212.·9-s + 486.·11-s − 428.·13-s + 1.80e3·17-s + 1.04e3·19-s − 380.·21-s − 686.·23-s − 2.51e3·27-s − 1.33e3·29-s − 7.99e3·31-s + 2.68e3·33-s − 1.97e3·37-s − 2.36e3·39-s + 1.07e4·41-s − 1.50e4·43-s − 895.·47-s − 1.20e4·49-s + 9.94e3·51-s − 1.93e4·53-s + 5.78e3·57-s − 2.11e4·59-s − 2.77e4·61-s + 1.46e4·63-s + 7.71e3·67-s − 3.79e3·69-s + ⋯
L(s)  = 1  + 0.354·3-s − 0.531·7-s − 0.874·9-s + 1.21·11-s − 0.703·13-s + 1.51·17-s + 0.665·19-s − 0.188·21-s − 0.270·23-s − 0.664·27-s − 0.295·29-s − 1.49·31-s + 0.429·33-s − 0.236·37-s − 0.249·39-s + 1.00·41-s − 1.23·43-s − 0.0591·47-s − 0.717·49-s + 0.535·51-s − 0.945·53-s + 0.235·57-s − 0.792·59-s − 0.953·61-s + 0.465·63-s + 0.210·67-s − 0.0959·69-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 64.153564.1535
Root analytic conductor: 8.009588.00958
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 400, ( :5/2), −1)(2,\ 400,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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)
bad2 1 1
5 1 1
good3 1−5.52T+243T2 1 - 5.52T + 243T^{2}
7 1+68.9T+1.68e4T2 1 + 68.9T + 1.68e4T^{2}
11 1−486.T+1.61e5T2 1 - 486.T + 1.61e5T^{2}
13 1+428.T+3.71e5T2 1 + 428.T + 3.71e5T^{2}
17 1−1.80e3T+1.41e6T2 1 - 1.80e3T + 1.41e6T^{2}
19 1−1.04e3T+2.47e6T2 1 - 1.04e3T + 2.47e6T^{2}
23 1+686.T+6.43e6T2 1 + 686.T + 6.43e6T^{2}
29 1+1.33e3T+2.05e7T2 1 + 1.33e3T + 2.05e7T^{2}
31 1+7.99e3T+2.86e7T2 1 + 7.99e3T + 2.86e7T^{2}
37 1+1.97e3T+6.93e7T2 1 + 1.97e3T + 6.93e7T^{2}
41 1−1.07e4T+1.15e8T2 1 - 1.07e4T + 1.15e8T^{2}
43 1+1.50e4T+1.47e8T2 1 + 1.50e4T + 1.47e8T^{2}
47 1+895.T+2.29e8T2 1 + 895.T + 2.29e8T^{2}
53 1+1.93e4T+4.18e8T2 1 + 1.93e4T + 4.18e8T^{2}
59 1+2.11e4T+7.14e8T2 1 + 2.11e4T + 7.14e8T^{2}
61 1+2.77e4T+8.44e8T2 1 + 2.77e4T + 8.44e8T^{2}
67 1−7.71e3T+1.35e9T2 1 - 7.71e3T + 1.35e9T^{2}
71 1−5.14e4T+1.80e9T2 1 - 5.14e4T + 1.80e9T^{2}
73 1+4.37e4T+2.07e9T2 1 + 4.37e4T + 2.07e9T^{2}
79 1−6.22e3T+3.07e9T2 1 - 6.22e3T + 3.07e9T^{2}
83 1+5.29e4T+3.93e9T2 1 + 5.29e4T + 3.93e9T^{2}
89 1−4.46e4T+5.58e9T2 1 - 4.46e4T + 5.58e9T^{2}
97 1+1.48e5T+8.58e9T2 1 + 1.48e5T + 8.58e9T^{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

−9.663760769952137382378764424603, −9.326389073826344049558828735669, −8.143048867060283140636510069865, −7.26543367833479084050704529738, −6.15034666101399144060676425011, −5.24612615091120063156159555819, −3.74478506992973619234538897316, −2.98033014974405254573320178382, −1.50412646747505076647988632968, 0, 1.50412646747505076647988632968, 2.98033014974405254573320178382, 3.74478506992973619234538897316, 5.24612615091120063156159555819, 6.15034666101399144060676425011, 7.26543367833479084050704529738, 8.143048867060283140636510069865, 9.326389073826344049558828735669, 9.663760769952137382378764424603

Graph of the ZZ-function along the critical line