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Properties

Label 2-1440-1.1-c3-0-12
Degree 22
Conductor 14401440
Sign 11
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 1.74·7-s − 28.9·11-s − 48.2·13-s − 16.2·17-s − 130.·19-s + 182.·23-s + 25·25-s − 291.·29-s + 219.·31-s + 8.73·35-s + 436.·37-s + 339.·41-s + 316.·43-s − 335.·47-s − 339.·49-s + 520.·53-s − 144.·55-s + 589.·59-s − 566.·61-s − 241.·65-s + 407.·67-s + 486.·71-s + 143.·73-s − 50.6·77-s + 968.·79-s − 532.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.0943·7-s − 0.794·11-s − 1.02·13-s − 0.231·17-s − 1.57·19-s + 1.65·23-s + 0.200·25-s − 1.86·29-s + 1.27·31-s + 0.0422·35-s + 1.93·37-s + 1.29·41-s + 1.12·43-s − 1.04·47-s − 0.991·49-s + 1.34·53-s − 0.355·55-s + 1.30·59-s − 1.18·61-s − 0.460·65-s + 0.742·67-s + 0.812·71-s + 0.229·73-s − 0.0749·77-s + 1.37·79-s − 0.704·83-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: 11
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1440, ( :3/2), 1)(2,\ 1440,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.8389184771.838918477
L(12)L(\frac12) ≈\approx 1.8389184771.838918477
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 1−5T 1 - 5T
good7 1−1.74T+343T2 1 - 1.74T + 343T^{2}
11 1+28.9T+1.33e3T2 1 + 28.9T + 1.33e3T^{2}
13 1+48.2T+2.19e3T2 1 + 48.2T + 2.19e3T^{2}
17 1+16.2T+4.91e3T2 1 + 16.2T + 4.91e3T^{2}
19 1+130.T+6.85e3T2 1 + 130.T + 6.85e3T^{2}
23 1−182.T+1.21e4T2 1 - 182.T + 1.21e4T^{2}
29 1+291.T+2.43e4T2 1 + 291.T + 2.43e4T^{2}
31 1−219.T+2.97e4T2 1 - 219.T + 2.97e4T^{2}
37 1−436.T+5.06e4T2 1 - 436.T + 5.06e4T^{2}
41 1−339.T+6.89e4T2 1 - 339.T + 6.89e4T^{2}
43 1−316.T+7.95e4T2 1 - 316.T + 7.95e4T^{2}
47 1+335.T+1.03e5T2 1 + 335.T + 1.03e5T^{2}
53 1−520.T+1.48e5T2 1 - 520.T + 1.48e5T^{2}
59 1−589.T+2.05e5T2 1 - 589.T + 2.05e5T^{2}
61 1+566.T+2.26e5T2 1 + 566.T + 2.26e5T^{2}
67 1−407.T+3.00e5T2 1 - 407.T + 3.00e5T^{2}
71 1−486.T+3.57e5T2 1 - 486.T + 3.57e5T^{2}
73 1−143.T+3.89e5T2 1 - 143.T + 3.89e5T^{2}
79 1−968.T+4.93e5T2 1 - 968.T + 4.93e5T^{2}
83 1+532.T+5.71e5T2 1 + 532.T + 5.71e5T^{2}
89 1+67.8T+7.04e5T2 1 + 67.8T + 7.04e5T^{2}
97 1−218.T+9.12e5T2 1 - 218.T + 9.12e5T^{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

−9.308512139555393232182662598428, −8.327354213251203685812491872786, −7.56927378934543411240532753381, −6.73298807393961043426792631396, −5.84036727135581625349582303971, −4.97389523542831345951528206394, −4.21224768658033662536849521249, −2.78005104036952292404722075943, −2.15220079774395696503825157241, −0.64710079732116236320918425889, 0.64710079732116236320918425889, 2.15220079774395696503825157241, 2.78005104036952292404722075943, 4.21224768658033662536849521249, 4.97389523542831345951528206394, 5.84036727135581625349582303971, 6.73298807393961043426792631396, 7.56927378934543411240532753381, 8.327354213251203685812491872786, 9.308512139555393232182662598428

Graph of the ZZ-function along the critical line