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Properties

Label 2-28e2-1.1-c5-0-25
Degree 22
Conductor 784784
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.9·3-s − 34.9·5-s − 19.1·9-s − 748.·11-s − 12.4·13-s − 523.·15-s + 2.21e3·17-s + 1.35e3·19-s − 3.02e3·23-s − 1.90e3·25-s − 3.92e3·27-s + 5.02e3·29-s − 4.55e3·31-s − 1.11e4·33-s − 1.04e3·37-s − 186.·39-s + 3.24e3·41-s + 2.86e3·43-s + 669.·45-s + 2.05e4·47-s + 3.31e4·51-s − 2.20e4·53-s + 2.61e4·55-s + 2.03e4·57-s − 1.88e4·59-s + 5.62e4·61-s + 436.·65-s + ⋯
L(s)  = 1  + 0.959·3-s − 0.625·5-s − 0.0786·9-s − 1.86·11-s − 0.0204·13-s − 0.600·15-s + 1.85·17-s + 0.862·19-s − 1.19·23-s − 0.608·25-s − 1.03·27-s + 1.10·29-s − 0.851·31-s − 1.78·33-s − 0.125·37-s − 0.0196·39-s + 0.301·41-s + 0.236·43-s + 0.0492·45-s + 1.35·47-s + 1.78·51-s − 1.07·53-s + 1.16·55-s + 0.827·57-s − 0.705·59-s + 1.93·61-s + 0.0128·65-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: 11
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: 00
Selberg data: (2, 784, ( :5/2), 1)(2,\ 784,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.0180147202.018014720
L(12)L(\frac12) ≈\approx 2.0180147202.018014720
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 1−14.9T+243T2 1 - 14.9T + 243T^{2}
5 1+34.9T+3.12e3T2 1 + 34.9T + 3.12e3T^{2}
11 1+748.T+1.61e5T2 1 + 748.T + 1.61e5T^{2}
13 1+12.4T+3.71e5T2 1 + 12.4T + 3.71e5T^{2}
17 1−2.21e3T+1.41e6T2 1 - 2.21e3T + 1.41e6T^{2}
19 1−1.35e3T+2.47e6T2 1 - 1.35e3T + 2.47e6T^{2}
23 1+3.02e3T+6.43e6T2 1 + 3.02e3T + 6.43e6T^{2}
29 1−5.02e3T+2.05e7T2 1 - 5.02e3T + 2.05e7T^{2}
31 1+4.55e3T+2.86e7T2 1 + 4.55e3T + 2.86e7T^{2}
37 1+1.04e3T+6.93e7T2 1 + 1.04e3T + 6.93e7T^{2}
41 1−3.24e3T+1.15e8T2 1 - 3.24e3T + 1.15e8T^{2}
43 1−2.86e3T+1.47e8T2 1 - 2.86e3T + 1.47e8T^{2}
47 1−2.05e4T+2.29e8T2 1 - 2.05e4T + 2.29e8T^{2}
53 1+2.20e4T+4.18e8T2 1 + 2.20e4T + 4.18e8T^{2}
59 1+1.88e4T+7.14e8T2 1 + 1.88e4T + 7.14e8T^{2}
61 1−5.62e4T+8.44e8T2 1 - 5.62e4T + 8.44e8T^{2}
67 1+1.40e4T+1.35e9T2 1 + 1.40e4T + 1.35e9T^{2}
71 1−7.44e4T+1.80e9T2 1 - 7.44e4T + 1.80e9T^{2}
73 1−3.96e4T+2.07e9T2 1 - 3.96e4T + 2.07e9T^{2}
79 1−3.91e4T+3.07e9T2 1 - 3.91e4T + 3.07e9T^{2}
83 1−1.36e4T+3.93e9T2 1 - 1.36e4T + 3.93e9T^{2}
89 1+6.72e4T+5.58e9T2 1 + 6.72e4T + 5.58e9T^{2}
97 1−8.87e4T+8.58e9T2 1 - 8.87e4T + 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

−9.612177553404070233440197304701, −8.407315953104697870785575093956, −7.82324353666045478060152913350, −7.50641602870707467503818495041, −5.84008824497677940850339044713, −5.14582210140878800703956235036, −3.76957749205374689146933615675, −3.06880480942162575780622449277, −2.16778901206258942046495967187, −0.59598900651526776494752887366, 0.59598900651526776494752887366, 2.16778901206258942046495967187, 3.06880480942162575780622449277, 3.76957749205374689146933615675, 5.14582210140878800703956235036, 5.84008824497677940850339044713, 7.50641602870707467503818495041, 7.82324353666045478060152913350, 8.407315953104697870785575093956, 9.612177553404070233440197304701

Graph of the ZZ-function along the critical line