↑

Properties

Label 2-2400-1.1-c1-0-34
Degree 22
Conductor 24002400
Sign −1-1
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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 + 9-s − 4·11-s − 2·13-s + 2·17-s − 8·19-s + 4·23-s + 27-s − 6·29-s − 4·33-s − 2·37-s − 2·39-s − 6·41-s + 4·43-s − 12·47-s − 7·49-s + 2·51-s + 6·53-s − 8·57-s − 12·59-s + 14·61-s − 12·67-s + 4·69-s − 2·73-s + 8·79-s + 81-s − 4·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 1.83·19-s + 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.280·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 1.79·61-s − 1.46·67-s + 0.481·69-s − 0.234·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2400, ( :1/2), −1)(2,\ 2400,\ (\ :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 1
good7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 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 1−2T+pT2 1 - 2 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.504660516100362643211307216113, −7.894696834542226361022801674411, −7.18138745760652059052193457513, −6.32654489351111017845946854020, −5.29471439982575242401030102022, −4.62185459688259708266199771078, −3.55388893647426173836027608466, −2.66987544017520648744955346644, −1.79780397316567120302158729682, 0, 1.79780397316567120302158729682, 2.66987544017520648744955346644, 3.55388893647426173836027608466, 4.62185459688259708266199771078, 5.29471439982575242401030102022, 6.32654489351111017845946854020, 7.18138745760652059052193457513, 7.894696834542226361022801674411, 8.504660516100362643211307216113

Graph of the ZZ-function along the critical line