Properties

Label 2-2400-1.1-c1-0-6
Degree 22
Conductor 24002400
Sign 11
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 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·7-s + 9-s − 4·13-s + 8·19-s − 4·21-s + 4·23-s + 27-s − 6·29-s + 8·31-s + 4·37-s − 4·39-s + 6·41-s + 4·43-s − 4·47-s + 9·49-s + 12·53-s + 8·57-s − 6·61-s − 4·63-s + 12·67-s + 4·69-s + 16·71-s + 8·79-s + 81-s − 12·83-s − 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.64·53-s + 1.05·57-s − 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s + 1.89·71-s + 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯

Functional equation

Λ(s)=(2400s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2400s/2ΓC(s+1/2)L(s)=(Λ(1s)\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    =    253522^{5} \cdot 3 \cdot 5^{2}
Sign: 11
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: 00
Selberg data: (2, 2400, ( :1/2), 1)(2,\ 2400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 1.7730162031.773016203
L(12)L(\frac12) \approx 1.7730162031.773016203
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1T 1 - T
5 1 1
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 18T+pT2 1 - 8 T + p T^{2}
23 14T+pT2 1 - 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 18T+pT2 1 - 8 T + p T^{2}
37 14T+pT2 1 - 4 T + p T^{2}
41 16T+pT2 1 - 6 T + p T^{2}
43 14T+pT2 1 - 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 112T+pT2 1 - 12 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 112T+pT2 1 - 12 T + p T^{2}
71 116T+pT2 1 - 16 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 18T+pT2 1 - 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−9.294439362155582311396720712775, −8.146688029147244337134377106094, −7.33074642876103566465615298847, −6.86435994056926016287184004578, −5.86257513923387218027991820847, −5.02892022270217661724574337422, −3.92685848464838036707491107183, −3.07713744288718387943470887456, −2.48768118964892092450282845760, −0.819834028977665625201711882906, 0.819834028977665625201711882906, 2.48768118964892092450282845760, 3.07713744288718387943470887456, 3.92685848464838036707491107183, 5.02892022270217661724574337422, 5.86257513923387218027991820847, 6.86435994056926016287184004578, 7.33074642876103566465615298847, 8.146688029147244337134377106094, 9.294439362155582311396720712775

Graph of the ZZ-function along the critical line