↑

Properties

Label 2-70e2-1.1-c1-0-56
Degree 22
Conductor 49004900
Sign −1-1
Analytic cond. 39.126639.1266
Root an. cond. 6.255136.25513
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 − 2·9-s + 6·11-s + 2·13-s − 6·17-s − 8·19-s − 3·23-s − 5·27-s + 3·29-s − 2·31-s + 6·33-s − 8·37-s + 2·39-s + 3·41-s − 5·43-s − 6·51-s − 12·53-s − 8·57-s + 61-s + 7·67-s − 3·69-s − 10·73-s − 4·79-s + 81-s + 3·83-s + 3·87-s + 3·89-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.80·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.625·23-s − 0.962·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s + 0.320·39-s + 0.468·41-s − 0.762·43-s − 0.840·51-s − 1.64·53-s − 1.05·57-s + 0.128·61-s + 0.855·67-s − 0.361·69-s − 1.17·73-s − 0.450·79-s + 1/9·81-s + 0.329·83-s + 0.321·87-s + 0.317·89-s + ⋯

Functional equation

Λ(s)=(4900s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4900s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49004900    =    22⋅52⋅722^{2} \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 39.126639.1266
Root analytic conductor: 6.255136.25513
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4900, ( :1/2), −1)(2,\ 4900,\ (\ :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
5 1 1
7 1 1
good3 1−T+pT2 1 - T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−8.211270859347756885349671452319, −7.04879334883551896766303355152, −6.40678059519075227049276802788, −6.03236251086300149239737522411, −4.73220950631722382800727147838, −4.04233006113911691325892171640, −3.43864198876714594107947372560, −2.30245811067245526829398566091, −1.60712841656694145068894267650, 0, 1.60712841656694145068894267650, 2.30245811067245526829398566091, 3.43864198876714594107947372560, 4.04233006113911691325892171640, 4.73220950631722382800727147838, 6.03236251086300149239737522411, 6.40678059519075227049276802788, 7.04879334883551896766303355152, 8.211270859347756885349671452319

Graph of the ZZ-function along the critical line