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Properties

Label 2-43560-1.1-c1-0-46
Degree 22
Conductor 4356043560
Sign −1-1
Analytic cond. 347.828347.828
Root an. cond. 18.650118.6501
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  − 5-s + 2·7-s − 4·13-s + 6·17-s − 2·19-s − 8·23-s + 25-s + 6·29-s − 2·35-s − 6·37-s + 10·41-s + 2·43-s − 12·47-s − 3·49-s + 6·53-s − 8·59-s + 4·65-s − 4·67-s + 12·71-s + 4·73-s + 10·79-s − 4·83-s − 6·85-s + 14·89-s − 8·91-s + 2·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s − 0.986·37-s + 1.56·41-s + 0.304·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s + 0.496·65-s − 0.488·67-s + 1.42·71-s + 0.468·73-s + 1.12·79-s − 0.439·83-s − 0.650·85-s + 1.48·89-s − 0.838·91-s + 0.205·95-s + 0.203·97-s + ⋯

Functional equation

Λ(s)=(43560s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4356043560    =    23⋅32⋅5⋅1122^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 347.828347.828
Root analytic conductor: 18.650118.6501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 43560, ( :1/2), −1)(2,\ 43560,\ (\ :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)
bad2 1 1
3 1 1
5 1+T 1 + T
11 1 1
good7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−14.75366340830658, −14.45836869885670, −14.10890529216813, −13.51423934857104, −12.54599962190332, −12.42075846101122, −11.82872936586909, −11.51144089964069, −10.55308940803512, −10.44175078080498, −9.647967515710248, −9.298630421945830, −8.304695453082943, −8.027469662162821, −7.685322454201581, −6.966177397381054, −6.319971163018528, −5.678615545440838, −5.006345893892092, −4.602431088626003, −3.889652774864658, −3.265359317109748, −2.458591700606634, −1.829060069984947, −0.9608456924655442, 0, 0.9608456924655442, 1.829060069984947, 2.458591700606634, 3.265359317109748, 3.889652774864658, 4.602431088626003, 5.006345893892092, 5.678615545440838, 6.319971163018528, 6.966177397381054, 7.685322454201581, 8.027469662162821, 8.304695453082943, 9.298630421945830, 9.647967515710248, 10.44175078080498, 10.55308940803512, 11.51144089964069, 11.82872936586909, 12.42075846101122, 12.54599962190332, 13.51423934857104, 14.10890529216813, 14.45836869885670, 14.75366340830658

Graph of the ZZ-function along the critical line