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Properties

Label 2-1440-1.1-c3-0-44
Degree 22
Conductor 14401440
Sign −1-1
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 12·7-s − 24·11-s + 38·13-s + 6·17-s − 104·19-s + 100·23-s + 25·25-s − 230·29-s + 56·31-s − 60·35-s + 190·37-s − 202·41-s + 148·43-s + 124·47-s − 199·49-s − 206·53-s + 120·55-s − 128·59-s + 190·61-s − 190·65-s + 204·67-s − 440·71-s + 1.21e3·73-s − 288·77-s − 816·79-s − 1.41e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.647·7-s − 0.657·11-s + 0.810·13-s + 0.0856·17-s − 1.25·19-s + 0.906·23-s + 1/5·25-s − 1.47·29-s + 0.324·31-s − 0.289·35-s + 0.844·37-s − 0.769·41-s + 0.524·43-s + 0.384·47-s − 0.580·49-s − 0.533·53-s + 0.294·55-s − 0.282·59-s + 0.398·61-s − 0.362·65-s + 0.371·67-s − 0.735·71-s + 1.93·73-s − 0.426·77-s − 1.16·79-s − 1.86·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1440, ( :3/2), −1)(2,\ 1440,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1
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

−8.596009101264268901615977740946, −8.052448932088165840640417275772, −7.24922944837136919963292779525, −6.29687729877037352317403189296, −5.37383114228527808554612771293, −4.50305967378684014932242907157, −3.64094120651138828383101231820, −2.49286487754356437406755043437, −1.33717700198834086459302718927, 0, 1.33717700198834086459302718927, 2.49286487754356437406755043437, 3.64094120651138828383101231820, 4.50305967378684014932242907157, 5.37383114228527808554612771293, 6.29687729877037352317403189296, 7.24922944837136919963292779525, 8.052448932088165840640417275772, 8.596009101264268901615977740946

Graph of the ZZ-function along the critical line