↑

Properties

Label 2-28e2-1.1-c5-0-9
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10.0·3-s − 79.8·5-s − 141.·9-s − 351.·11-s − 291.·13-s − 804.·15-s − 370.·17-s − 1.50e3·19-s + 425.·23-s + 3.24e3·25-s − 3.87e3·27-s − 7.78e3·29-s + 2.57e3·31-s − 3.54e3·33-s + 739.·37-s − 2.94e3·39-s + 7.02e3·41-s − 1.83e3·43-s + 1.12e4·45-s + 1.53e3·47-s − 3.73e3·51-s − 9.53e3·53-s + 2.80e4·55-s − 1.51e4·57-s + 2.96e4·59-s − 4.65e4·61-s + 2.32e4·65-s + ⋯
L(s)  = 1  + 0.646·3-s − 1.42·5-s − 0.581·9-s − 0.876·11-s − 0.478·13-s − 0.923·15-s − 0.310·17-s − 0.956·19-s + 0.167·23-s + 1.03·25-s − 1.02·27-s − 1.71·29-s + 0.481·31-s − 0.567·33-s + 0.0888·37-s − 0.309·39-s + 0.653·41-s − 0.151·43-s + 0.830·45-s + 0.101·47-s − 0.200·51-s − 0.466·53-s + 1.25·55-s − 0.618·57-s + 1.10·59-s − 1.60·61-s + 0.683·65-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 0.60594019070.6059401907
L(12)L(\frac12) ≈\approx 0.60594019070.6059401907
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 1−10.0T+243T2 1 - 10.0T + 243T^{2}
5 1+79.8T+3.12e3T2 1 + 79.8T + 3.12e3T^{2}
11 1+351.T+1.61e5T2 1 + 351.T + 1.61e5T^{2}
13 1+291.T+3.71e5T2 1 + 291.T + 3.71e5T^{2}
17 1+370.T+1.41e6T2 1 + 370.T + 1.41e6T^{2}
19 1+1.50e3T+2.47e6T2 1 + 1.50e3T + 2.47e6T^{2}
23 1−425.T+6.43e6T2 1 - 425.T + 6.43e6T^{2}
29 1+7.78e3T+2.05e7T2 1 + 7.78e3T + 2.05e7T^{2}
31 1−2.57e3T+2.86e7T2 1 - 2.57e3T + 2.86e7T^{2}
37 1−739.T+6.93e7T2 1 - 739.T + 6.93e7T^{2}
41 1−7.02e3T+1.15e8T2 1 - 7.02e3T + 1.15e8T^{2}
43 1+1.83e3T+1.47e8T2 1 + 1.83e3T + 1.47e8T^{2}
47 1−1.53e3T+2.29e8T2 1 - 1.53e3T + 2.29e8T^{2}
53 1+9.53e3T+4.18e8T2 1 + 9.53e3T + 4.18e8T^{2}
59 1−2.96e4T+7.14e8T2 1 - 2.96e4T + 7.14e8T^{2}
61 1+4.65e4T+8.44e8T2 1 + 4.65e4T + 8.44e8T^{2}
67 1+2.67e4T+1.35e9T2 1 + 2.67e4T + 1.35e9T^{2}
71 1−1.43e4T+1.80e9T2 1 - 1.43e4T + 1.80e9T^{2}
73 1+7.00e4T+2.07e9T2 1 + 7.00e4T + 2.07e9T^{2}
79 1−2.70e4T+3.07e9T2 1 - 2.70e4T + 3.07e9T^{2}
83 1−7.97e4T+3.93e9T2 1 - 7.97e4T + 3.93e9T^{2}
89 1−4.35e4T+5.58e9T2 1 - 4.35e4T + 5.58e9T^{2}
97 1−1.03e5T+8.58e9T2 1 - 1.03e5T + 8.58e9T^{2}
show more
show less
   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

−9.307759976446065559328091921899, −8.564817453295444105363838353724, −7.82165641951233527777299216094, −7.35001764832547198476923126388, −6.04555844584081343154050568686, −4.88358503213123938750922381228, −3.96323708230849900041991978065, −3.08286700135097381506353677487, −2.14233959711887185248407727140, −0.32223496566032188068650867867, 0.32223496566032188068650867867, 2.14233959711887185248407727140, 3.08286700135097381506353677487, 3.96323708230849900041991978065, 4.88358503213123938750922381228, 6.04555844584081343154050568686, 7.35001764832547198476923126388, 7.82165641951233527777299216094, 8.564817453295444105363838353724, 9.307759976446065559328091921899

Graph of the ZZ-function along the critical line