↑

Properties

Label 2-1872-1.1-c3-0-80
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  + 7·5-s + 13·7-s − 26·11-s + 13·13-s − 77·17-s + 126·19-s − 96·23-s − 76·25-s + 82·29-s − 196·31-s + 91·35-s − 131·37-s − 336·41-s + 201·43-s − 105·47-s − 174·49-s + 432·53-s − 182·55-s − 294·59-s − 56·61-s + 91·65-s − 478·67-s + 9·71-s + 98·73-s − 338·77-s − 1.30e3·79-s − 308·83-s + ⋯
L(s)  = 1  + 0.626·5-s + 0.701·7-s − 0.712·11-s + 0.277·13-s − 1.09·17-s + 1.52·19-s − 0.870·23-s − 0.607·25-s + 0.525·29-s − 1.13·31-s + 0.439·35-s − 0.582·37-s − 1.27·41-s + 0.712·43-s − 0.325·47-s − 0.507·49-s + 1.11·53-s − 0.446·55-s − 0.648·59-s − 0.117·61-s + 0.173·65-s − 0.871·67-s + 0.0150·71-s + 0.157·73-s − 0.500·77-s − 1.85·79-s − 0.407·83-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 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
7 1−13T+p3T2 1 - 13 T + p^{3} T^{2}
11 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
17 1+77T+p3T2 1 + 77 T + p^{3} T^{2}
19 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
23 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
29 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
31 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
37 1+131T+p3T2 1 + 131 T + p^{3} T^{2}
41 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
43 1−201T+p3T2 1 - 201 T + p^{3} T^{2}
47 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
53 1−432T+p3T2 1 - 432 T + p^{3} T^{2}
59 1+294T+p3T2 1 + 294 T + p^{3} T^{2}
61 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
67 1+478T+p3T2 1 + 478 T + p^{3} T^{2}
71 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
73 1−98T+p3T2 1 - 98 T + p^{3} T^{2}
79 1+1304T+p3T2 1 + 1304 T + p^{3} T^{2}
83 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
89 1−1190T+p3T2 1 - 1190 T + p^{3} T^{2}
97 1−70T+p3T2 1 - 70 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

−8.480086513468905640578713219306, −7.72213909623913587015032488339, −6.96721111218392620606767480054, −5.91727302272227481488266734350, −5.31636048711874553081784215114, −4.47356408088371820942110661105, −3.36481540098648956502320318929, −2.25163826307861076138383214535, −1.45884912883188450472695849084, 0, 1.45884912883188450472695849084, 2.25163826307861076138383214535, 3.36481540098648956502320318929, 4.47356408088371820942110661105, 5.31636048711874553081784215114, 5.91727302272227481488266734350, 6.96721111218392620606767480054, 7.72213909623913587015032488339, 8.480086513468905640578713219306

Graph of the ZZ-function along the critical line