↑

Properties

Label 2-1872-1.1-c3-0-76
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
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 + 26·7-s − 52·11-s − 13·13-s + 48·17-s − 18·19-s + 52·23-s − 121·25-s + 224·29-s − 310·31-s + 52·35-s − 18·37-s + 330·41-s − 328·43-s − 616·47-s + 333·49-s − 324·53-s − 104·55-s − 188·59-s − 110·61-s − 26·65-s − 118·67-s + 656·71-s − 178·73-s − 1.35e3·77-s − 836·79-s − 60·83-s + ⋯
L(s)  = 1  + 0.178·5-s + 1.40·7-s − 1.42·11-s − 0.277·13-s + 0.684·17-s − 0.217·19-s + 0.471·23-s − 0.967·25-s + 1.43·29-s − 1.79·31-s + 0.251·35-s − 0.0799·37-s + 1.25·41-s − 1.16·43-s − 1.91·47-s + 0.970·49-s − 0.839·53-s − 0.254·55-s − 0.414·59-s − 0.230·61-s − 0.0496·65-s − 0.215·67-s + 1.09·71-s − 0.285·73-s − 2.00·77-s − 1.19·79-s − 0.0793·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :3/2), −1)(2,\ 1872,\ (\ :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 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
11 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
17 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
19 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
23 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
29 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
31 1+10pT+p3T2 1 + 10 p T + p^{3} T^{2}
37 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
41 1−330T+p3T2 1 - 330 T + p^{3} T^{2}
43 1+328T+p3T2 1 + 328 T + p^{3} T^{2}
47 1+616T+p3T2 1 + 616 T + p^{3} T^{2}
53 1+324T+p3T2 1 + 324 T + p^{3} T^{2}
59 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
61 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
67 1+118T+p3T2 1 + 118 T + p^{3} T^{2}
71 1−656T+p3T2 1 - 656 T + p^{3} T^{2}
73 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
79 1+836T+p3T2 1 + 836 T + p^{3} T^{2}
83 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
89 1−870T+p3T2 1 - 870 T + p^{3} T^{2}
97 1−1238T+p3T2 1 - 1238 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

−8.135012405650642115749201203539, −7.967842107324843502944554482522, −7.06958064319213035349027766732, −5.89049158878586314574295723891, −5.13110231626004154442641865849, −4.63191182902429422519003391810, −3.33402360311650353828879734588, −2.29154885505320094380682933342, −1.41463155805051003515867229203, 0, 1.41463155805051003515867229203, 2.29154885505320094380682933342, 3.33402360311650353828879734588, 4.63191182902429422519003391810, 5.13110231626004154442641865849, 5.89049158878586314574295723891, 7.06958064319213035349027766732, 7.967842107324843502944554482522, 8.135012405650642115749201203539

Graph of the ZZ-function along the critical line