↑

Properties

Label 2-43560-1.1-c1-0-11
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 4·13-s + 2·17-s + 2·19-s − 4·23-s + 25-s + 6·29-s − 2·35-s + 10·37-s + 2·41-s + 2·43-s − 3·49-s − 6·53-s − 4·61-s − 4·65-s + 4·67-s + 2·79-s + 2·85-s − 10·89-s + 8·91-s + 2·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 1.10·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.512·61-s − 0.496·65-s + 0.488·67-s + 0.225·79-s + 0.216·85-s − 1.05·89-s + 0.838·91-s + 0.205·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 1.8049933371.804993337
L(12)L(\frac12) ≈\approx 1.8049933371.804993337
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 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+pT2 1 + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+10T+pT2 1 + 10 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

−14.63102578078452, −14.02775356187380, −13.86705754379050, −12.98530034635506, −12.66223123044243, −12.21143477543739, −11.62338576263863, −11.06560140910586, −10.29537763070177, −9.891229242381357, −9.591091986590180, −9.046980979777885, −8.223869292039187, −7.759200076466167, −7.199779482286860, −6.507297193710076, −6.088608188988365, −5.494587098072641, −4.794076194843055, −4.283311567245118, −3.423268865117186, −2.810163407938494, −2.316884992921861, −1.369591431472191, −0.4924981658695197, 0.4924981658695197, 1.369591431472191, 2.316884992921861, 2.810163407938494, 3.423268865117186, 4.283311567245118, 4.794076194843055, 5.494587098072641, 6.088608188988365, 6.507297193710076, 7.199779482286860, 7.759200076466167, 8.223869292039187, 9.046980979777885, 9.591091986590180, 9.891229242381357, 10.29537763070177, 11.06560140910586, 11.62338576263863, 12.21143477543739, 12.66223123044243, 12.98530034635506, 13.86705754379050, 14.02775356187380, 14.63102578078452

Graph of the ZZ-function along the critical line