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Properties

Label 2-16245-1.1-c1-0-9
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 + 3·11-s + 6·13-s + 4·14-s − 4·16-s − 6·17-s + 2·20-s − 6·22-s + 8·23-s + 25-s − 12·26-s − 4·28-s − 7·29-s + 9·31-s + 8·32-s + 12·34-s − 2·35-s − 2·37-s − 6·41-s + 10·43-s + 6·44-s − 16·46-s − 4·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s − 0.632·10-s + 0.904·11-s + 1.66·13-s + 1.06·14-s − 16-s − 1.45·17-s + 0.447·20-s − 1.27·22-s + 1.66·23-s + 1/5·25-s − 2.35·26-s − 0.755·28-s − 1.29·29-s + 1.61·31-s + 1.41·32-s + 2.05·34-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.904·44-s − 2.35·46-s − 0.583·47-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 1−3T+pT2 1 - 3 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+2T+pT2 1 + 2 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+4T+pT2 1 + 4 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+3T+pT2 1 + 3 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−16.34282933144755, −15.78559287572106, −15.45834936958880, −14.69836767548134, −13.79811183769072, −13.44391658520617, −13.04929326117170, −12.25834003626368, −11.32007050572292, −11.02412150903052, −10.65074962290063, −9.711776686360789, −9.365027847329194, −8.899155653931670, −8.502414365771857, −7.735372608527110, −6.867539628814433, −6.502792736488725, −6.113504249696727, −4.985152346187800, −4.244078580201647, −3.440346219583338, −2.641239497506407, −1.607860427970451, −1.123835423275056, 0, 1.123835423275056, 1.607860427970451, 2.641239497506407, 3.440346219583338, 4.244078580201647, 4.985152346187800, 6.113504249696727, 6.502792736488725, 6.867539628814433, 7.735372608527110, 8.502414365771857, 8.899155653931670, 9.365027847329194, 9.711776686360789, 10.65074962290063, 11.02412150903052, 11.32007050572292, 12.25834003626368, 13.04929326117170, 13.44391658520617, 13.79811183769072, 14.69836767548134, 15.45834936958880, 15.78559287572106, 16.34282933144755

Graph of the ZZ-function along the critical line