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Properties

Label 2-89280-1.1-c1-0-58
Degree 22
Conductor 8928089280
Sign 11
Analytic cond. 712.904712.904
Root an. cond. 26.700226.7002
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s + 3·11-s + 4·13-s + 6·17-s + 19-s + 3·23-s + 25-s + 31-s + 35-s + 4·37-s + 43-s + 12·47-s − 6·49-s + 9·53-s − 3·55-s + 10·61-s − 4·65-s + 10·67-s − 15·71-s − 7·73-s − 3·77-s + 5·79-s − 6·83-s − 6·85-s − 15·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.904·11-s + 1.10·13-s + 1.45·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s + 0.179·31-s + 0.169·35-s + 0.657·37-s + 0.152·43-s + 1.75·47-s − 6/7·49-s + 1.23·53-s − 0.404·55-s + 1.28·61-s − 0.496·65-s + 1.22·67-s − 1.78·71-s − 0.819·73-s − 0.341·77-s + 0.562·79-s − 0.658·83-s − 0.650·85-s − 1.58·89-s − 0.419·91-s + ⋯

Functional equation

Λ(s)=(89280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(89280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8928089280    =    26⋅32⋅5⋅312^{6} \cdot 3^{2} \cdot 5 \cdot 31
Sign: 11
Analytic conductor: 712.904712.904
Root analytic conductor: 26.700226.7002
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 89280, ( :1/2), 1)(2,\ 89280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3557425703.355742570
L(12)L(\frac12) ≈\approx 3.3557425703.355742570
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
31 1−T 1 - T
good7 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+15T+pT2 1 + 15 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−13.94590847913856, −13.40036984405540, −12.79268645433368, −12.44572328460261, −11.88331658566213, −11.33561381861593, −11.15046333999812, −10.23791359107491, −10.02221450945494, −9.342612904917263, −8.809075767937565, −8.445879522061161, −7.814217530113987, −7.198246787074151, −6.873075940384310, −6.082169052297953, −5.759590638744122, −5.167712475028388, −4.213647150302554, −4.007818933337258, −3.246721276249019, −2.924381850668159, −1.892320462107714, −1.071077285250344, −0.7219853824307759, 0.7219853824307759, 1.071077285250344, 1.892320462107714, 2.924381850668159, 3.246721276249019, 4.007818933337258, 4.213647150302554, 5.167712475028388, 5.759590638744122, 6.082169052297953, 6.873075940384310, 7.198246787074151, 7.814217530113987, 8.445879522061161, 8.809075767937565, 9.342612904917263, 10.02221450945494, 10.23791359107491, 11.15046333999812, 11.33561381861593, 11.88331658566213, 12.44572328460261, 12.79268645433368, 13.40036984405540, 13.94590847913856

Graph of the ZZ-function along the critical line