↑

Properties

Label 2-468-1.1-c3-0-11
Degree 22
Conductor 468468
Sign −1-1
Analytic cond. 27.612827.6128
Root an. cond. 5.254795.25479
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 32·7-s + 68·11-s + 13·13-s + 14·17-s + 4·19-s − 72·23-s − 121·25-s − 102·29-s − 136·31-s − 64·35-s − 386·37-s − 250·41-s − 140·43-s + 296·47-s + 681·49-s − 526·53-s + 136·55-s − 332·59-s − 410·61-s + 26·65-s + 596·67-s + 880·71-s + 506·73-s − 2.17e3·77-s − 640·79-s − 1.38e3·83-s + ⋯
L(s)  = 1  + 0.178·5-s − 1.72·7-s + 1.86·11-s + 0.277·13-s + 0.199·17-s + 0.0482·19-s − 0.652·23-s − 0.967·25-s − 0.653·29-s − 0.787·31-s − 0.309·35-s − 1.71·37-s − 0.952·41-s − 0.496·43-s + 0.918·47-s + 1.98·49-s − 1.36·53-s + 0.333·55-s − 0.732·59-s − 0.860·61-s + 0.0496·65-s + 1.08·67-s + 1.47·71-s + 0.811·73-s − 3.22·77-s − 0.911·79-s − 1.82·83-s + ⋯

Functional equation

Λ(s)=(468s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(468s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 468468    =    22⋅32⋅132^{2} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 27.612827.6128
Root analytic conductor: 5.254795.25479
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 468, ( :3/2), −1)(2,\ 468,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
13 1−pT 1 - p T
good5 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
7 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
11 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
17 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
19 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1+102T+p3T2 1 + 102 T + p^{3} T^{2}
31 1+136T+p3T2 1 + 136 T + p^{3} T^{2}
37 1+386T+p3T2 1 + 386 T + p^{3} T^{2}
41 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
43 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
47 1−296T+p3T2 1 - 296 T + p^{3} T^{2}
53 1+526T+p3T2 1 + 526 T + p^{3} T^{2}
59 1+332T+p3T2 1 + 332 T + p^{3} T^{2}
61 1+410T+p3T2 1 + 410 T + p^{3} T^{2}
67 1−596T+p3T2 1 - 596 T + p^{3} T^{2}
71 1−880T+p3T2 1 - 880 T + p^{3} T^{2}
73 1−506T+p3T2 1 - 506 T + p^{3} T^{2}
79 1+640T+p3T2 1 + 640 T + p^{3} T^{2}
83 1+1380T+p3T2 1 + 1380 T + p^{3} T^{2}
89 1+1450T+p3T2 1 + 1450 T + p^{3} T^{2}
97 1+446T+p3T2 1 + 446 T + p^{3} 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

−9.854874963591984556567134082862, −9.476546513981004135339123064073, −8.579754041476501043358351941633, −7.12115902380027228718903012808, −6.45933825527282320912357798157, −5.68120483184912008031639667316, −3.96021769021079225511584685798, −3.35692263862781988931026014923, −1.67059934989288439502634804305, 0, 1.67059934989288439502634804305, 3.35692263862781988931026014923, 3.96021769021079225511584685798, 5.68120483184912008031639667316, 6.45933825527282320912357798157, 7.12115902380027228718903012808, 8.579754041476501043358351941633, 9.476546513981004135339123064073, 9.854874963591984556567134082862

Graph of the ZZ-function along the critical line