↑

Properties

Label 2-80-1.1-c7-0-12
Degree 22
Conductor 8080
Sign −1-1
Analytic cond. 24.990824.9908
Root an. cond. 4.999084.99908
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 36·3-s + 125·5-s − 776·7-s − 891·9-s + 124·11-s − 1.30e4·13-s + 4.50e3·15-s − 1.59e4·17-s + 2.05e4·19-s − 2.79e4·21-s + 2.92e4·23-s + 1.56e4·25-s − 1.10e5·27-s − 2.21e5·29-s + 1.09e5·31-s + 4.46e3·33-s − 9.70e4·35-s + 7.34e4·37-s − 4.70e5·39-s + 1.27e4·41-s − 2.90e5·43-s − 1.11e5·45-s − 1.26e6·47-s − 2.21e5·49-s − 5.74e5·51-s − 3.95e5·53-s + 1.55e4·55-s + ⋯
L(s)  = 1  + 0.769·3-s + 0.447·5-s − 0.855·7-s − 0.407·9-s + 0.0280·11-s − 1.65·13-s + 0.344·15-s − 0.787·17-s + 0.686·19-s − 0.658·21-s + 0.500·23-s + 1/5·25-s − 1.08·27-s − 1.68·29-s + 0.661·31-s + 0.0216·33-s − 0.382·35-s + 0.238·37-s − 1.27·39-s + 0.0289·41-s − 0.557·43-s − 0.182·45-s − 1.78·47-s − 0.268·49-s − 0.606·51-s − 0.365·53-s + 0.0125·55-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −1-1
Analytic conductor: 24.990824.9908
Root analytic conductor: 4.999084.99908
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 80, ( :7/2), −1)(2,\ 80,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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
5 1−p3T 1 - p^{3} T
good3 1−4p2T+p7T2 1 - 4 p^{2} T + p^{7} T^{2}
7 1+776T+p7T2 1 + 776 T + p^{7} T^{2}
11 1−124T+p7T2 1 - 124 T + p^{7} T^{2}
13 1+13082T+p7T2 1 + 13082 T + p^{7} T^{2}
17 1+15950T+p7T2 1 + 15950 T + p^{7} T^{2}
19 1−20516T+p7T2 1 - 20516 T + p^{7} T^{2}
23 1−29224T+p7T2 1 - 29224 T + p^{7} T^{2}
29 1+221482T+p7T2 1 + 221482 T + p^{7} T^{2}
31 1−109760T+p7T2 1 - 109760 T + p^{7} T^{2}
37 1−73422T+p7T2 1 - 73422 T + p^{7} T^{2}
41 1−12762T+p7T2 1 - 12762 T + p^{7} T^{2}
43 1+290548T+p7T2 1 + 290548 T + p^{7} T^{2}
47 1+1269152T+p7T2 1 + 1269152 T + p^{7} T^{2}
53 1+395778T+p7T2 1 + 395778 T + p^{7} T^{2}
59 1+421492T+p7T2 1 + 421492 T + p^{7} T^{2}
61 1+2122250T+p7T2 1 + 2122250 T + p^{7} T^{2}
67 1−3132868T+p7T2 1 - 3132868 T + p^{7} T^{2}
71 1−5376552T+p7T2 1 - 5376552 T + p^{7} T^{2}
73 1−4985466T+p7T2 1 - 4985466 T + p^{7} T^{2}
79 1+3867504T+p7T2 1 + 3867504 T + p^{7} T^{2}
83 1−6190196T+p7T2 1 - 6190196 T + p^{7} T^{2}
89 1−1124394T+p7T2 1 - 1124394 T + p^{7} T^{2}
97 1−9968098T+p7T2 1 - 9968098 T + p^{7} 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

−12.64807893876384547387021669959, −11.33023705772055506535114603085, −9.774292927455621135157787189598, −9.256811001897131496001900092966, −7.82758963286784406788785511872, −6.59144689519885099851781754421, −5.10356500691533893872930739944, −3.29366578871756918340465239756, −2.21565093762334056604785333535, 0, 2.21565093762334056604785333535, 3.29366578871756918340465239756, 5.10356500691533893872930739944, 6.59144689519885099851781754421, 7.82758963286784406788785511872, 9.256811001897131496001900092966, 9.774292927455621135157787189598, 11.33023705772055506535114603085, 12.64807893876384547387021669959

Graph of the ZZ-function along the critical line