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Properties

Label 2-1440-1.1-c3-0-15
Degree 22
Conductor 14401440
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 32·7-s − 64·11-s − 6·13-s − 38·17-s − 116·19-s + 120·23-s + 25·25-s + 122·29-s + 164·31-s − 160·35-s + 146·37-s + 238·41-s − 148·43-s + 184·47-s + 681·49-s − 470·53-s + 320·55-s + 216·59-s + 806·61-s + 30·65-s − 732·67-s − 264·71-s − 638·73-s − 2.04e3·77-s + 596·79-s + 884·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.72·7-s − 1.75·11-s − 0.128·13-s − 0.542·17-s − 1.40·19-s + 1.08·23-s + 1/5·25-s + 0.781·29-s + 0.950·31-s − 0.772·35-s + 0.648·37-s + 0.906·41-s − 0.524·43-s + 0.571·47-s + 1.98·49-s − 1.21·53-s + 0.784·55-s + 0.476·59-s + 1.69·61-s + 0.0572·65-s − 1.33·67-s − 0.441·71-s − 1.02·73-s − 3.03·77-s + 0.848·79-s + 1.16·83-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: yes
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 1.9396673901.939667390
L(12)L(\frac12) ≈\approx 1.9396673901.939667390
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 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
11 1+64T+p3T2 1 + 64 T + p^{3} T^{2}
13 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
17 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
19 1+116T+p3T2 1 + 116 T + p^{3} T^{2}
23 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
29 1−122T+p3T2 1 - 122 T + p^{3} T^{2}
31 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
37 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
41 1−238T+p3T2 1 - 238 T + p^{3} T^{2}
43 1+148T+p3T2 1 + 148 T + p^{3} T^{2}
47 1−184T+p3T2 1 - 184 T + p^{3} T^{2}
53 1+470T+p3T2 1 + 470 T + p^{3} T^{2}
59 1−216T+p3T2 1 - 216 T + p^{3} T^{2}
61 1−806T+p3T2 1 - 806 T + p^{3} T^{2}
67 1+732T+p3T2 1 + 732 T + p^{3} T^{2}
71 1+264T+p3T2 1 + 264 T + p^{3} T^{2}
73 1+638T+p3T2 1 + 638 T + p^{3} T^{2}
79 1−596T+p3T2 1 - 596 T + p^{3} T^{2}
83 1−884T+p3T2 1 - 884 T + p^{3} T^{2}
89 1+930T+p3T2 1 + 930 T + p^{3} T^{2}
97 1−322T+p3T2 1 - 322 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

−8.870017360566454134041475571094, −8.222865535898623116303426649794, −7.79898643419207128714246271461, −6.88299930433256774026635223295, −5.69765821472405700922559400059, −4.74018066909448217879659390512, −4.45795335143521183959730068488, −2.84718238724257255418807461441, −2.04211362598052834391328688738, −0.67913846168074242709035281313, 0.67913846168074242709035281313, 2.04211362598052834391328688738, 2.84718238724257255418807461441, 4.45795335143521183959730068488, 4.74018066909448217879659390512, 5.69765821472405700922559400059, 6.88299930433256774026635223295, 7.79898643419207128714246271461, 8.222865535898623116303426649794, 8.870017360566454134041475571094

Graph of the ZZ-function along the critical line