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Properties

Label 2-43560-1.1-c1-0-10
Degree 22
Conductor 4356043560
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 6·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s − 4·35-s − 6·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s − 6·61-s + 6·65-s − 4·67-s + 14·73-s − 16·79-s + 12·83-s − 2·85-s − 2·89-s − 24·91-s − 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.676·35-s − 0.986·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s − 0.768·61-s + 0.744·65-s − 0.488·67-s + 1.63·73-s − 1.80·79-s + 1.31·83-s − 0.216·85-s − 0.211·89-s − 2.51·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-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: 11
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: 00
Selberg data: (2, 43560, ( :1/2), 1)(2,\ 43560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7336883201.733688320
L(12)L(\frac12) ≈\approx 1.7336883201.733688320
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+4T+pT2 1 + 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 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 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+2T+pT2 1 + 2 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

−14.72954504162295, −14.07174704085523, −13.41185730757061, −13.20957095941046, −12.70902222491310, −12.40285813451298, −11.34285968428663, −10.93186379157511, −10.68954234479197, −9.885787302415063, −9.262380539919902, −9.085720699191505, −8.485191135442812, −7.759516299488151, −6.884254725741490, −6.669849703329678, −5.992046520740455, −5.720056708159651, −4.774365817793668, −4.112478417781127, −3.375095972852491, −3.078921701875804, −2.165850716709027, −1.388482685029479, −0.4832094432165871, 0.4832094432165871, 1.388482685029479, 2.165850716709027, 3.078921701875804, 3.375095972852491, 4.112478417781127, 4.774365817793668, 5.720056708159651, 5.992046520740455, 6.669849703329678, 6.884254725741490, 7.759516299488151, 8.485191135442812, 9.085720699191505, 9.262380539919902, 9.885787302415063, 10.68954234479197, 10.93186379157511, 11.34285968428663, 12.40285813451298, 12.70902222491310, 13.20957095941046, 13.41185730757061, 14.07174704085523, 14.72954504162295

Graph of the ZZ-function along the critical line