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Properties

Label 2-504-1.1-c5-0-15
Degree 22
Conductor 504504
Sign 11
Analytic cond. 80.833480.8334
Root an. cond. 8.990748.99074
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 64·5-s + 49·7-s + 54·11-s + 738·13-s + 848·17-s − 1.60e3·19-s + 3.67e3·23-s + 971·25-s + 4.33e3·29-s − 4.76e3·31-s + 3.13e3·35-s − 2.09e3·37-s + 6.11e3·41-s + 7.91e3·43-s − 6.57e3·47-s + 2.40e3·49-s + 7.89e3·53-s + 3.45e3·55-s + 4.16e4·59-s − 2.65e4·61-s + 4.72e4·65-s − 4.17e4·67-s − 8.35e4·71-s − 4.23e4·73-s + 2.64e3·77-s + 508·79-s + 8.36e3·83-s + ⋯
L(s)  = 1  + 1.14·5-s + 0.377·7-s + 0.134·11-s + 1.21·13-s + 0.711·17-s − 1.01·19-s + 1.44·23-s + 0.310·25-s + 0.956·29-s − 0.889·31-s + 0.432·35-s − 0.251·37-s + 0.568·41-s + 0.652·43-s − 0.433·47-s + 1/7·49-s + 0.386·53-s + 0.154·55-s + 1.55·59-s − 0.914·61-s + 1.38·65-s − 1.13·67-s − 1.96·71-s − 0.929·73-s + 0.0508·77-s + 0.00915·79-s + 0.133·83-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: yes
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 3.3129999653.312999965
L(12)L(\frac12) ≈\approx 3.3129999653.312999965
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−p2T 1 - p^{2} T
good5 1−64T+p5T2 1 - 64 T + p^{5} T^{2}
11 1−54T+p5T2 1 - 54 T + p^{5} T^{2}
13 1−738T+p5T2 1 - 738 T + p^{5} T^{2}
17 1−848T+p5T2 1 - 848 T + p^{5} T^{2}
19 1+1604T+p5T2 1 + 1604 T + p^{5} T^{2}
23 1−3670T+p5T2 1 - 3670 T + p^{5} T^{2}
29 1−4330T+p5T2 1 - 4330 T + p^{5} T^{2}
31 1+4760T+p5T2 1 + 4760 T + p^{5} T^{2}
37 1+2094T+p5T2 1 + 2094 T + p^{5} T^{2}
41 1−6116T+p5T2 1 - 6116 T + p^{5} T^{2}
43 1−7916T+p5T2 1 - 7916 T + p^{5} T^{2}
47 1+6572T+p5T2 1 + 6572 T + p^{5} T^{2}
53 1−7894T+p5T2 1 - 7894 T + p^{5} T^{2}
59 1−41664T+p5T2 1 - 41664 T + p^{5} T^{2}
61 1+26570T+p5T2 1 + 26570 T + p^{5} T^{2}
67 1+41736T+p5T2 1 + 41736 T + p^{5} T^{2}
71 1+83574T+p5T2 1 + 83574 T + p^{5} T^{2}
73 1+42314T+p5T2 1 + 42314 T + p^{5} T^{2}
79 1−508T+p5T2 1 - 508 T + p^{5} T^{2}
83 1−8364T+p5T2 1 - 8364 T + p^{5} T^{2}
89 1−49220T+p5T2 1 - 49220 T + p^{5} T^{2}
97 1−159670T+p5T2 1 - 159670 T + p^{5} T^{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

−10.24755236305674278114788987020, −9.121108480003522196397689721717, −8.582201559138271856077053649963, −7.34470859964643354675025009506, −6.26497883438430669857089961687, −5.61961693622016325626816009064, −4.48699761885991409843363556175, −3.18800116347132288058836707451, −1.93021999561154536816148043662, −0.983854113433832782894183770802, 0.983854113433832782894183770802, 1.93021999561154536816148043662, 3.18800116347132288058836707451, 4.48699761885991409843363556175, 5.61961693622016325626816009064, 6.26497883438430669857089961687, 7.34470859964643354675025009506, 8.582201559138271856077053649963, 9.121108480003522196397689721717, 10.24755236305674278114788987020

Graph of the ZZ-function along the critical line