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Properties

Label 2-6480-1.1-c1-0-79
Degree 22
Conductor 64806480
Sign −1-1
Analytic cond. 51.743051.7430
Root an. cond. 7.193267.19326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 5·11-s + 3·17-s − 5·19-s − 6·23-s + 25-s − 10·29-s + 2·31-s + 4·37-s − 3·41-s − 3·43-s − 4·47-s − 7·49-s − 6·53-s − 5·55-s + 3·59-s + 2·61-s + 11·67-s + 14·71-s − 15·73-s − 10·79-s + 12·83-s − 3·85-s + 14·89-s + 5·95-s − 13·97-s − 12·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.727·17-s − 1.14·19-s − 1.25·23-s + 1/5·25-s − 1.85·29-s + 0.359·31-s + 0.657·37-s − 0.468·41-s − 0.457·43-s − 0.583·47-s − 49-s − 0.824·53-s − 0.674·55-s + 0.390·59-s + 0.256·61-s + 1.34·67-s + 1.66·71-s − 1.75·73-s − 1.12·79-s + 1.31·83-s − 0.325·85-s + 1.48·89-s + 0.512·95-s − 1.31·97-s − 1.19·101-s + ⋯

Functional equation

Λ(s)=(6480s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6480s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 64806480    =    24⋅34⋅52^{4} \cdot 3^{4} \cdot 5
Sign: −1-1
Analytic conductor: 51.743051.7430
Root analytic conductor: 7.193267.19326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6480, ( :1/2), −1)(2,\ 6480,\ (\ :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 1
5 1+T 1 + T
good7 1+pT2 1 + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1+3T+pT2 1 + 3 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1−14T+pT2 1 - 14 T + p T^{2}
73 1+15T+pT2 1 + 15 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+13T+pT2 1 + 13 T + p T^{2}
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   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

−7.81212256184125416581208696729, −6.84030052849159377245471985233, −6.36315767456006162547057716195, −5.62281187732482829008822990279, −4.64605279824741366722167485179, −3.89307308908582402198992282959, −3.47412284263825399275488195861, −2.17888958965834404694640156988, −1.35261254883323916209184071170, 0, 1.35261254883323916209184071170, 2.17888958965834404694640156988, 3.47412284263825399275488195861, 3.89307308908582402198992282959, 4.64605279824741366722167485179, 5.62281187732482829008822990279, 6.36315767456006162547057716195, 6.84030052849159377245471985233, 7.81212256184125416581208696729

Graph of the ZZ-function along the critical line