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Properties

Label 2-28e2-1.1-c5-0-46
Degree 22
Conductor 784784
Sign −1-1
Analytic cond. 125.740125.740
Root an. cond. 11.213411.2134
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  − 25.6·3-s − 28.7·5-s + 414.·9-s + 270.·11-s − 300.·13-s + 737.·15-s − 613.·17-s − 1.70e3·19-s − 3.18e3·23-s − 2.29e3·25-s − 4.40e3·27-s + 4.29e3·29-s + 2.02e3·31-s − 6.92e3·33-s + 5.15e3·37-s + 7.71e3·39-s + 7.14e3·41-s + 1.95e4·43-s − 1.19e4·45-s + 1.99e4·47-s + 1.57e4·51-s + 3.94e3·53-s − 7.76e3·55-s + 4.36e4·57-s − 2.97e4·59-s + 5.05e4·61-s + 8.64e3·65-s + ⋯
L(s)  = 1  − 1.64·3-s − 0.514·5-s + 1.70·9-s + 0.673·11-s − 0.493·13-s + 0.846·15-s − 0.514·17-s − 1.08·19-s − 1.25·23-s − 0.735·25-s − 1.16·27-s + 0.949·29-s + 0.379·31-s − 1.10·33-s + 0.618·37-s + 0.811·39-s + 0.663·41-s + 1.61·43-s − 0.878·45-s + 1.32·47-s + 0.846·51-s + 0.193·53-s − 0.346·55-s + 1.77·57-s − 1.11·59-s + 1.73·61-s + 0.253·65-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 125.740125.740
Root analytic conductor: 11.213411.2134
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 784, ( :5/2), −1)(2,\ 784,\ (\ :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
7 1 1
good3 1+25.6T+243T2 1 + 25.6T + 243T^{2}
5 1+28.7T+3.12e3T2 1 + 28.7T + 3.12e3T^{2}
11 1−270.T+1.61e5T2 1 - 270.T + 1.61e5T^{2}
13 1+300.T+3.71e5T2 1 + 300.T + 3.71e5T^{2}
17 1+613.T+1.41e6T2 1 + 613.T + 1.41e6T^{2}
19 1+1.70e3T+2.47e6T2 1 + 1.70e3T + 2.47e6T^{2}
23 1+3.18e3T+6.43e6T2 1 + 3.18e3T + 6.43e6T^{2}
29 1−4.29e3T+2.05e7T2 1 - 4.29e3T + 2.05e7T^{2}
31 1−2.02e3T+2.86e7T2 1 - 2.02e3T + 2.86e7T^{2}
37 1−5.15e3T+6.93e7T2 1 - 5.15e3T + 6.93e7T^{2}
41 1−7.14e3T+1.15e8T2 1 - 7.14e3T + 1.15e8T^{2}
43 1−1.95e4T+1.47e8T2 1 - 1.95e4T + 1.47e8T^{2}
47 1−1.99e4T+2.29e8T2 1 - 1.99e4T + 2.29e8T^{2}
53 1−3.94e3T+4.18e8T2 1 - 3.94e3T + 4.18e8T^{2}
59 1+2.97e4T+7.14e8T2 1 + 2.97e4T + 7.14e8T^{2}
61 1−5.05e4T+8.44e8T2 1 - 5.05e4T + 8.44e8T^{2}
67 1+5.05e3T+1.35e9T2 1 + 5.05e3T + 1.35e9T^{2}
71 1+3.28e4T+1.80e9T2 1 + 3.28e4T + 1.80e9T^{2}
73 1−1.11e4T+2.07e9T2 1 - 1.11e4T + 2.07e9T^{2}
79 1+8.18e4T+3.07e9T2 1 + 8.18e4T + 3.07e9T^{2}
83 1−1.18e5T+3.93e9T2 1 - 1.18e5T + 3.93e9T^{2}
89 1−4.16e4T+5.58e9T2 1 - 4.16e4T + 5.58e9T^{2}
97 1+4.36e4T+8.58e9T2 1 + 4.36e4T + 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

−9.268534523482326171955663128888, −8.121340049307635827705013313140, −7.16710195815937774521569249382, −6.31241041136095772976598767488, −5.75690042585202149226125922469, −4.49346847522710412689369171761, −4.09069814619518151244127467890, −2.27166446622554228351210149928, −0.886221142913795246061684425110, 0, 0.886221142913795246061684425110, 2.27166446622554228351210149928, 4.09069814619518151244127467890, 4.49346847522710412689369171761, 5.75690042585202149226125922469, 6.31241041136095772976598767488, 7.16710195815937774521569249382, 8.121340049307635827705013313140, 9.268534523482326171955663128888

Graph of the ZZ-function along the critical line