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Properties

Label 2-16245-1.1-c1-0-12
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  + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s − 11-s + 2·13-s − 4·14-s − 4·16-s − 2·17-s + 2·20-s − 2·22-s + 4·23-s + 25-s + 4·26-s − 4·28-s + 5·29-s + 9·31-s − 8·32-s − 4·34-s − 2·35-s − 6·37-s − 6·41-s − 10·43-s − 2·44-s + 8·46-s − 3·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.301·11-s + 0.554·13-s − 1.06·14-s − 16-s − 0.485·17-s + 0.447·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s + 0.928·29-s + 1.61·31-s − 1.41·32-s − 0.685·34-s − 0.338·35-s − 0.986·37-s − 0.937·41-s − 1.52·43-s − 0.301·44-s + 1.17·46-s − 3/7·49-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−pT+pT2 1 - p T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+7T+pT2 1 + 7 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−15.88916821064774, −15.49568639494830, −15.22820948132853, −14.34394402837085, −13.87682315461369, −13.46491456378255, −13.02760572883278, −12.55405963888388, −11.88438441663771, −11.44269228474228, −10.65808195401795, −10.10281234128630, −9.533204285210794, −8.660498527260020, −8.402303343010062, −7.178960331395390, −6.654950678715853, −6.251723949669223, −5.616524913783952, −4.836086910142212, −4.523888741426811, −3.449824645287124, −3.115748464352751, −2.395648367665589, −1.360015620797932, 0, 1.360015620797932, 2.395648367665589, 3.115748464352751, 3.449824645287124, 4.523888741426811, 4.836086910142212, 5.616524913783952, 6.251723949669223, 6.654950678715853, 7.178960331395390, 8.402303343010062, 8.660498527260020, 9.533204285210794, 10.10281234128630, 10.65808195401795, 11.44269228474228, 11.88438441663771, 12.55405963888388, 13.02760572883278, 13.46491456378255, 13.87682315461369, 14.34394402837085, 15.22820948132853, 15.49568639494830, 15.88916821064774

Graph of the ZZ-function along the critical line