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Properties

Label 2-504-1.1-c5-0-8
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  − 32·5-s + 49·7-s + 624·11-s − 708·13-s − 934·17-s + 1.85e3·19-s + 1.12e3·23-s − 2.10e3·25-s + 1.17e3·29-s + 2.90e3·31-s − 1.56e3·35-s − 1.24e4·37-s − 2.66e3·41-s − 7.14e3·43-s + 7.46e3·47-s + 2.40e3·49-s + 2.72e4·53-s − 1.99e4·55-s − 2.49e3·59-s − 1.10e4·61-s + 2.26e4·65-s + 3.97e4·67-s + 6.98e4·71-s + 1.64e4·73-s + 3.05e4·77-s + 7.83e4·79-s − 1.09e5·83-s + ⋯
L(s)  = 1  − 0.572·5-s + 0.377·7-s + 1.55·11-s − 1.16·13-s − 0.783·17-s + 1.18·19-s + 0.441·23-s − 0.672·25-s + 0.259·29-s + 0.543·31-s − 0.216·35-s − 1.49·37-s − 0.247·41-s − 0.589·43-s + 0.493·47-s + 1/7·49-s + 1.33·53-s − 0.890·55-s − 0.0931·59-s − 0.381·61-s + 0.665·65-s + 1.08·67-s + 1.64·71-s + 0.361·73-s + 0.587·77-s + 1.41·79-s − 1.74·83-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 1.9538017371.953801737
L(12)L(\frac12) ≈\approx 1.9538017371.953801737
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+32T+p5T2 1 + 32 T + p^{5} T^{2}
11 1−624T+p5T2 1 - 624 T + p^{5} T^{2}
13 1+708T+p5T2 1 + 708 T + p^{5} T^{2}
17 1+934T+p5T2 1 + 934 T + p^{5} T^{2}
19 1−1858T+p5T2 1 - 1858 T + p^{5} T^{2}
23 1−1120T+p5T2 1 - 1120 T + p^{5} T^{2}
29 1−1174T+p5T2 1 - 1174 T + p^{5} T^{2}
31 1−2908T+p5T2 1 - 2908 T + p^{5} T^{2}
37 1+12462T+p5T2 1 + 12462 T + p^{5} T^{2}
41 1+2662T+p5T2 1 + 2662 T + p^{5} T^{2}
43 1+7144T+p5T2 1 + 7144 T + p^{5} T^{2}
47 1−7468T+p5T2 1 - 7468 T + p^{5} T^{2}
53 1−27274T+p5T2 1 - 27274 T + p^{5} T^{2}
59 1+2490T+p5T2 1 + 2490 T + p^{5} T^{2}
61 1+11096T+p5T2 1 + 11096 T + p^{5} T^{2}
67 1−39756T+p5T2 1 - 39756 T + p^{5} T^{2}
71 1−69888T+p5T2 1 - 69888 T + p^{5} T^{2}
73 1−16450T+p5T2 1 - 16450 T + p^{5} T^{2}
79 1−78376T+p5T2 1 - 78376 T + p^{5} T^{2}
83 1+109818T+p5T2 1 + 109818 T + p^{5} T^{2}
89 1−56966T+p5T2 1 - 56966 T + p^{5} T^{2}
97 1+115946T+p5T2 1 + 115946 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

−10.02440519219289023206574267284, −9.235959471566366891246428240311, −8.366020585771334664440492652131, −7.30807815794708318943142528503, −6.67876116197668066964064355277, −5.31419901906406817022227176274, −4.37150364697475188852319536812, −3.40657596135082735799816824954, −1.99074323515603502775507146385, −0.71155608190346149353994044686, 0.71155608190346149353994044686, 1.99074323515603502775507146385, 3.40657596135082735799816824954, 4.37150364697475188852319536812, 5.31419901906406817022227176274, 6.67876116197668066964064355277, 7.30807815794708318943142528503, 8.366020585771334664440492652131, 9.235959471566366891246428240311, 10.02440519219289023206574267284

Graph of the ZZ-function along the critical line