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Properties

Label 2-16245-1.1-c1-0-10
Degree 22
Conductor 1624516245
Sign −1-1
Analytic cond. 129.716129.716
Root an. cond. 11.389311.3893
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s + 2·7-s + 3·11-s + 4·13-s + 4·16-s + 2·20-s + 6·23-s + 25-s − 4·28-s + 3·29-s − 5·31-s − 2·35-s − 8·37-s − 6·41-s − 4·43-s − 6·44-s − 6·47-s − 3·49-s − 8·52-s − 6·53-s − 3·55-s − 9·59-s − 7·61-s − 8·64-s − 4·65-s − 2·67-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s + 0.755·7-s + 0.904·11-s + 1.10·13-s + 16-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.755·28-s + 0.557·29-s − 0.898·31-s − 0.338·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.875·47-s − 3/7·49-s − 1.10·52-s − 0.824·53-s − 0.404·55-s − 1.17·59-s − 0.896·61-s − 64-s − 0.496·65-s − 0.244·67-s + ⋯

Functional equation

Λ(s)=(16245s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(16245s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1624516245    =    32⋅5⋅1923^{2} \cdot 5 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 129.716129.716
Root analytic conductor: 11.389311.3893
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 16245, ( :1/2), −1)(2,\ 16245,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad3 1 1
5 1+T 1 + T
19 1 1
good2 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−7T+pT2 1 - 7 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−16.24829612810809, −15.63334310445578, −14.86813195076504, −14.68151000234201, −13.96789764479291, −13.52058928395821, −12.99253657647404, −12.27989656472158, −11.80573222586068, −11.13023482981508, −10.70603162170148, −10.00126661624895, −9.066072187159856, −8.957344399867594, −8.277149390922542, −7.777426039744632, −6.939666790693225, −6.342751220658681, −5.516633157935133, −4.815609230056924, −4.453717839069794, −3.462008683487843, −3.293612713999307, −1.660432324832200, −1.206162012622432, 0, 1.206162012622432, 1.660432324832200, 3.293612713999307, 3.462008683487843, 4.453717839069794, 4.815609230056924, 5.516633157935133, 6.342751220658681, 6.939666790693225, 7.777426039744632, 8.277149390922542, 8.957344399867594, 9.066072187159856, 10.00126661624895, 10.70603162170148, 11.13023482981508, 11.80573222586068, 12.27989656472158, 12.99253657647404, 13.52058928395821, 13.96789764479291, 14.68151000234201, 14.86813195076504, 15.63334310445578, 16.24829612810809

Graph of the ZZ-function along the critical line