↑

Properties

Label 2-43560-1.1-c1-0-50
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·7-s + 4·13-s + 3·17-s + 3·19-s − 4·23-s + 25-s − 29-s − 5·31-s − 3·35-s − 5·37-s − 2·41-s + 8·43-s + 10·47-s + 2·49-s − 11·53-s − 61-s + 4·65-s + 4·67-s − 5·71-s − 10·73-s + 8·79-s + 3·85-s + 15·89-s − 12·91-s + 3·95-s − 6·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s + 1.10·13-s + 0.727·17-s + 0.688·19-s − 0.834·23-s + 1/5·25-s − 0.185·29-s − 0.898·31-s − 0.507·35-s − 0.821·37-s − 0.312·41-s + 1.21·43-s + 1.45·47-s + 2/7·49-s − 1.51·53-s − 0.128·61-s + 0.496·65-s + 0.488·67-s − 0.593·71-s − 1.17·73-s + 0.900·79-s + 0.325·85-s + 1.58·89-s − 1.25·91-s + 0.307·95-s − 0.609·97-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+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1+11T+pT2 1 + 11 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1+6T+pT2 1 + 6 T + p T^{2}
show more
show less
   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

−14.91305538931588, −14.29710305781303, −13.82979588981345, −13.48329425915663, −12.86996175661649, −12.40224576414506, −11.99079208751805, −11.22440990971415, −10.70696526882224, −10.21107244285676, −9.691315318787765, −9.146071987991753, −8.831050629175733, −7.919428399685221, −7.545237768583745, −6.750685380883449, −6.324193646083742, −5.672390248026672, −5.436123553857335, −4.403667793215739, −3.684785182209089, −3.328767997964070, −2.599313996665477, −1.727811153241631, −1.015083169927311, 0, 1.015083169927311, 1.727811153241631, 2.599313996665477, 3.328767997964070, 3.684785182209089, 4.403667793215739, 5.436123553857335, 5.672390248026672, 6.324193646083742, 6.750685380883449, 7.545237768583745, 7.919428399685221, 8.831050629175733, 9.146071987991753, 9.691315318787765, 10.21107244285676, 10.70696526882224, 11.22440990971415, 11.99079208751805, 12.40224576414506, 12.86996175661649, 13.48329425915663, 13.82979588981345, 14.29710305781303, 14.91305538931588

Graph of the ZZ-function along the critical line