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Properties

Label 2-89280-1.1-c1-0-75
Degree 22
Conductor 8928089280
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 2·11-s + 2·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s + 4·29-s − 31-s + 4·35-s − 2·37-s − 4·43-s + 12·47-s + 9·49-s + 6·53-s − 2·55-s + 8·59-s − 14·61-s − 2·65-s − 2·67-s − 10·71-s − 8·77-s − 4·79-s + 2·85-s + 6·89-s − 8·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s − 0.179·31-s + 0.676·35-s − 0.328·37-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s + 1.04·59-s − 1.79·61-s − 0.248·65-s − 0.244·67-s − 1.18·71-s − 0.911·77-s − 0.450·79-s + 0.216·85-s + 0.635·89-s − 0.838·91-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: −1-1
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: 11
Selberg data: (2, 89280, ( :1/2), −1)(2,\ 89280,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+4T+pT2 1 + 4 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 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 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−13.96621477653938, −13.54017218057608, −13.12815971142902, −12.68269782496640, −12.19427867707558, −11.76653625473745, −11.07941567014075, −10.61942973759075, −10.28616666389442, −9.526845069408157, −9.033418649542571, −8.785762912229314, −8.224286531640344, −7.330007404725857, −6.948338374827599, −6.553635126834483, −6.051533474209351, −5.452486720473217, −4.653743953201009, −4.090711282417015, −3.651828956804413, −2.951848674566388, −2.587988107007430, −1.532558822458179, −0.7853655760547264, 0, 0.7853655760547264, 1.532558822458179, 2.587988107007430, 2.951848674566388, 3.651828956804413, 4.090711282417015, 4.653743953201009, 5.452486720473217, 6.051533474209351, 6.553635126834483, 6.948338374827599, 7.330007404725857, 8.224286531640344, 8.785762912229314, 9.033418649542571, 9.526845069408157, 10.28616666389442, 10.61942973759075, 11.07941567014075, 11.76653625473745, 12.19427867707558, 12.68269782496640, 13.12815971142902, 13.54017218057608, 13.96621477653938

Graph of the ZZ-function along the critical line