↑

Properties

Label 2-3840-1.1-c1-0-33
Degree 22
Conductor 38403840
Sign −1-1
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 2·13-s + 15-s + 6·19-s + 4·21-s − 2·23-s + 25-s − 27-s − 10·29-s + 8·31-s − 4·33-s + 4·35-s − 6·37-s + 2·39-s + 2·41-s − 45-s + 10·47-s + 9·49-s + 6·53-s − 4·55-s − 6·57-s − 4·61-s − 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 1.37·19-s + 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s + 1.45·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.794·57-s − 0.512·61-s − 0.503·63-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3840, ( :1/2), −1)(2,\ 3840,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
5 1+T 1 + T
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1−10T+pT2 1 - 10 T + 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+12T+pT2 1 + 12 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+18T+pT2 1 + 18 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

−7.992883565045267623959574889367, −7.14171916920406481852093221362, −6.74376258764267807585390205531, −5.92663151211549582068628632033, −5.25940207031282311952344051521, −4.08213967612376426073032732169, −3.61521951901471238783945328372, −2.62673584549065035587996041061, −1.17018441156581271676063282459, 0, 1.17018441156581271676063282459, 2.62673584549065035587996041061, 3.61521951901471238783945328372, 4.08213967612376426073032732169, 5.25940207031282311952344051521, 5.92663151211549582068628632033, 6.74376258764267807585390205531, 7.14171916920406481852093221362, 7.992883565045267623959574889367

Graph of the ZZ-function along the critical line