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Properties

Label 2-480-1.1-c3-0-9
Degree 22
Conductor 480480
Sign 11
Analytic cond. 28.320928.3209
Root an. cond. 5.321745.32174
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·5-s − 12·7-s + 9·9-s − 24·11-s + 38·13-s + 15·15-s − 6·17-s + 104·19-s − 36·21-s + 100·23-s + 25·25-s + 27·27-s + 230·29-s − 56·31-s − 72·33-s − 60·35-s + 190·37-s + 114·39-s + 202·41-s − 148·43-s + 45·45-s + 124·47-s − 199·49-s − 18·51-s + 206·53-s − 120·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.647·7-s + 1/3·9-s − 0.657·11-s + 0.810·13-s + 0.258·15-s − 0.0856·17-s + 1.25·19-s − 0.374·21-s + 0.906·23-s + 1/5·25-s + 0.192·27-s + 1.47·29-s − 0.324·31-s − 0.379·33-s − 0.289·35-s + 0.844·37-s + 0.468·39-s + 0.769·41-s − 0.524·43-s + 0.149·45-s + 0.384·47-s − 0.580·49-s − 0.0494·51-s + 0.533·53-s − 0.294·55-s + ⋯

Functional equation

Λ(s)=(480s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(480s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 480480    =    25⋅3⋅52^{5} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 28.320928.3209
Root analytic conductor: 5.321745.32174
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 480, ( :3/2), 1)(2,\ 480,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5584643322.558464332
L(12)L(\frac12) ≈\approx 2.5584643322.558464332
L(52)L(\frac{5}{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−pT 1 - p T
5 1−pT 1 - p T
good7 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
11 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
13 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
17 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
19 1−104T+p3T2 1 - 104 T + p^{3} T^{2}
23 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
29 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
31 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
37 1−190T+p3T2 1 - 190 T + p^{3} T^{2}
41 1−202T+p3T2 1 - 202 T + p^{3} T^{2}
43 1+148T+p3T2 1 + 148 T + p^{3} T^{2}
47 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
53 1−206T+p3T2 1 - 206 T + p^{3} T^{2}
59 1+128T+p3T2 1 + 128 T + p^{3} T^{2}
61 1−190T+p3T2 1 - 190 T + p^{3} T^{2}
67 1+204T+p3T2 1 + 204 T + p^{3} T^{2}
71 1+440T+p3T2 1 + 440 T + p^{3} T^{2}
73 1−1210T+p3T2 1 - 1210 T + p^{3} T^{2}
79 1−816T+p3T2 1 - 816 T + p^{3} T^{2}
83 1+1412T+p3T2 1 + 1412 T + p^{3} T^{2}
89 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
97 1−1202T+p3T2 1 - 1202 T + p^{3} 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

−10.41622749054050663593390516684, −9.641939411294639579404550347131, −8.876707176376890709899324776573, −7.921304657796372278118400262727, −6.92189992030718392676078820046, −5.95122812575480884998599674005, −4.85378246152528777255751462926, −3.45768690804603567561083791554, −2.59956268418527990492689575188, −1.02473300525567783514756788390, 1.02473300525567783514756788390, 2.59956268418527990492689575188, 3.45768690804603567561083791554, 4.85378246152528777255751462926, 5.95122812575480884998599674005, 6.92189992030718392676078820046, 7.921304657796372278118400262727, 8.876707176376890709899324776573, 9.641939411294639579404550347131, 10.41622749054050663593390516684

Graph of the ZZ-function along the critical line