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Properties

Label 2-504-1.1-c5-0-14
Degree 22
Conductor 504504
Sign 11
Analytic cond. 80.833480.8334
Root an. cond. 8.990748.99074
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  + 97.5·5-s + 49·7-s − 406.·11-s − 905.·13-s − 359.·17-s + 1.81e3·19-s + 2.16e3·23-s + 6.38e3·25-s + 4.09e3·29-s + 4.45e3·31-s + 4.77e3·35-s + 8.93e3·37-s + 3.41e3·41-s − 8.69e3·43-s − 1.42e4·47-s + 2.40e3·49-s + 1.46e4·53-s − 3.96e4·55-s − 2.13e4·59-s + 5.40e4·61-s − 8.83e4·65-s + 4.49e4·67-s − 5.70e3·71-s + 4.76e4·73-s − 1.99e4·77-s + 5.80e3·79-s − 2.70e4·83-s + ⋯
L(s)  = 1  + 1.74·5-s + 0.377·7-s − 1.01·11-s − 1.48·13-s − 0.301·17-s + 1.15·19-s + 0.852·23-s + 2.04·25-s + 0.903·29-s + 0.831·31-s + 0.659·35-s + 1.07·37-s + 0.317·41-s − 0.717·43-s − 0.937·47-s + 0.142·49-s + 0.714·53-s − 1.76·55-s − 0.800·59-s + 1.86·61-s − 2.59·65-s + 1.22·67-s − 0.134·71-s + 1.04·73-s − 0.382·77-s + 0.104·79-s − 0.431·83-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 80.833480.8334
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 504, ( :5/2), 1)(2,\ 504,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.0795094583.079509458
L(12)L(\frac12) ≈\approx 3.0795094583.079509458
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
3 1 1
7 1−49T 1 - 49T
good5 1−97.5T+3.12e3T2 1 - 97.5T + 3.12e3T^{2}
11 1+406.T+1.61e5T2 1 + 406.T + 1.61e5T^{2}
13 1+905.T+3.71e5T2 1 + 905.T + 3.71e5T^{2}
17 1+359.T+1.41e6T2 1 + 359.T + 1.41e6T^{2}
19 1−1.81e3T+2.47e6T2 1 - 1.81e3T + 2.47e6T^{2}
23 1−2.16e3T+6.43e6T2 1 - 2.16e3T + 6.43e6T^{2}
29 1−4.09e3T+2.05e7T2 1 - 4.09e3T + 2.05e7T^{2}
31 1−4.45e3T+2.86e7T2 1 - 4.45e3T + 2.86e7T^{2}
37 1−8.93e3T+6.93e7T2 1 - 8.93e3T + 6.93e7T^{2}
41 1−3.41e3T+1.15e8T2 1 - 3.41e3T + 1.15e8T^{2}
43 1+8.69e3T+1.47e8T2 1 + 8.69e3T + 1.47e8T^{2}
47 1+1.42e4T+2.29e8T2 1 + 1.42e4T + 2.29e8T^{2}
53 1−1.46e4T+4.18e8T2 1 - 1.46e4T + 4.18e8T^{2}
59 1+2.13e4T+7.14e8T2 1 + 2.13e4T + 7.14e8T^{2}
61 1−5.40e4T+8.44e8T2 1 - 5.40e4T + 8.44e8T^{2}
67 1−4.49e4T+1.35e9T2 1 - 4.49e4T + 1.35e9T^{2}
71 1+5.70e3T+1.80e9T2 1 + 5.70e3T + 1.80e9T^{2}
73 1−4.76e4T+2.07e9T2 1 - 4.76e4T + 2.07e9T^{2}
79 1−5.80e3T+3.07e9T2 1 - 5.80e3T + 3.07e9T^{2}
83 1+2.70e4T+3.93e9T2 1 + 2.70e4T + 3.93e9T^{2}
89 1−9.62e4T+5.58e9T2 1 - 9.62e4T + 5.58e9T^{2}
97 1−1.01e5T+8.58e9T2 1 - 1.01e5T + 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

−9.898991370242585629651535132176, −9.591656256696332292364270270137, −8.388355494805211027420834492546, −7.33852326907518346972394024383, −6.38970651874797197063000701809, −5.24261944571308483278222942633, −4.90453838959611915338473952714, −2.84273132491704866433304713472, −2.20794676708162849652633382074, −0.897039847065639093826117081168, 0.897039847065639093826117081168, 2.20794676708162849652633382074, 2.84273132491704866433304713472, 4.90453838959611915338473952714, 5.24261944571308483278222942633, 6.38970651874797197063000701809, 7.33852326907518346972394024383, 8.388355494805211027420834492546, 9.591656256696332292364270270137, 9.898991370242585629651535132176

Graph of the ZZ-function along the critical line