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Properties

Label 2-9072-1.1-c1-0-119
Degree 22
Conductor 90729072
Sign −1-1
Analytic cond. 72.440272.4402
Root an. cond. 8.511188.51118
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  + 3·5-s − 7-s − 6·11-s + 2·13-s − 6·17-s + 7·19-s + 3·23-s + 4·25-s − 6·29-s − 2·31-s − 3·35-s + 2·37-s − 2·43-s + 49-s − 6·53-s − 18·55-s + 5·61-s + 6·65-s − 8·67-s + 3·71-s + 2·73-s + 6·77-s − 5·79-s + 12·83-s − 18·85-s − 2·91-s + 21·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 1.60·19-s + 0.625·23-s + 4/5·25-s − 1.11·29-s − 0.359·31-s − 0.507·35-s + 0.328·37-s − 0.304·43-s + 1/7·49-s − 0.824·53-s − 2.42·55-s + 0.640·61-s + 0.744·65-s − 0.977·67-s + 0.356·71-s + 0.234·73-s + 0.683·77-s − 0.562·79-s + 1.31·83-s − 1.95·85-s − 0.209·91-s + 2.15·95-s + ⋯

Functional equation

Λ(s)=(9072s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9072s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90729072    =    24⋅34⋅72^{4} \cdot 3^{4} \cdot 7
Sign: −1-1
Analytic conductor: 72.440272.4402
Root analytic conductor: 8.511188.51118
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9072, ( :1/2), −1)(2,\ 9072,\ (\ :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
7 1+T 1 + T
good5 1−3T+pT2 1 - 3 T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+5T+pT2 1 + 5 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−2T+pT2 1 - 2 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

−7.38476819275535999135405240527, −6.62509384532045389466627231804, −5.94865854409924744021641850334, −5.31974849755602342170659656944, −4.92966912255533521295876588044, −3.73180535825492880643668007815, −2.83036173067638134574426817684, −2.30854670098554420017510854630, −1.36002060033091265869350334541, 0, 1.36002060033091265869350334541, 2.30854670098554420017510854630, 2.83036173067638134574426817684, 3.73180535825492880643668007815, 4.92966912255533521295876588044, 5.31974849755602342170659656944, 5.94865854409924744021641850334, 6.62509384532045389466627231804, 7.38476819275535999135405240527

Graph of the ZZ-function along the critical line