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Properties

Label 2-89280-1.1-c1-0-51
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 − 2·7-s + 6·11-s + 4·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s + 2·29-s + 31-s + 2·35-s + 8·37-s − 8·41-s − 3·49-s + 6·53-s − 6·55-s − 6·59-s − 10·61-s − 4·65-s + 8·67-s + 10·73-s − 12·77-s + 4·83-s − 6·85-s − 8·91-s + 4·95-s + 18·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.80·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.179·31-s + 0.338·35-s + 1.31·37-s − 1.24·41-s − 3/7·49-s + 0.824·53-s − 0.809·55-s − 0.781·59-s − 1.28·61-s − 0.496·65-s + 0.977·67-s + 1.17·73-s − 1.36·77-s + 0.439·83-s − 0.650·85-s − 0.838·91-s + 0.410·95-s + 1.82·97-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 3.2631606073.263160607
L(12)L(\frac12) ≈\approx 3.2631606073.263160607
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+2T+pT2 1 + 2 T + p T^{2}
11 1−6T+pT2 1 - 6 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+4T+pT2 1 + 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−18T+pT2 1 - 18 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

−13.89255783106353, −13.31817609687713, −12.89676063791215, −12.34104650022038, −11.96879125134793, −11.38296105105809, −11.04251733667256, −10.38924586378829, −9.882673867754673, −9.242467632460937, −8.982457376323677, −8.402239514068716, −7.876049029940205, −7.170888399617107, −6.646345071804026, −6.291471318193739, −5.853783008249889, −4.967909744495364, −4.445024029016205, −3.695877667278979, −3.450632882571195, −2.908642724057448, −1.849252100426376, −1.108315091209293, −0.6970852214645254, 0.6970852214645254, 1.108315091209293, 1.849252100426376, 2.908642724057448, 3.450632882571195, 3.695877667278979, 4.445024029016205, 4.967909744495364, 5.853783008249889, 6.291471318193739, 6.646345071804026, 7.170888399617107, 7.876049029940205, 8.402239514068716, 8.982457376323677, 9.242467632460937, 9.882673867754673, 10.38924586378829, 11.04251733667256, 11.38296105105809, 11.96879125134793, 12.34104650022038, 12.89676063791215, 13.31817609687713, 13.89255783106353

Graph of the ZZ-function along the critical line