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Properties

Label 2-4140-1.1-c1-0-23
Degree 22
Conductor 41404140
Sign −1-1
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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 − 3·7-s + 2·11-s − 2·13-s + 7·17-s − 6·19-s − 23-s + 25-s + 9·29-s + 9·31-s + 3·35-s − 7·37-s − 5·41-s − 8·47-s + 2·49-s + 11·53-s − 2·55-s − 9·59-s + 2·65-s − 3·67-s − 3·71-s − 6·73-s − 6·77-s − 8·79-s − 5·83-s − 7·85-s + 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 0.603·11-s − 0.554·13-s + 1.69·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 1.67·29-s + 1.61·31-s + 0.507·35-s − 1.15·37-s − 0.780·41-s − 1.16·47-s + 2/7·49-s + 1.51·53-s − 0.269·55-s − 1.17·59-s + 0.248·65-s − 0.366·67-s − 0.356·71-s − 0.702·73-s − 0.683·77-s − 0.900·79-s − 0.548·83-s − 0.759·85-s + 0.628·91-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4140, ( :1/2), −1)(2,\ 4140,\ (\ :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
23 1+T 1 + T
good7 1+3T+pT2 1 + 3 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 1−7T+pT2 1 - 7 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+3T+pT2 1 + 3 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+5T+pT2 1 + 5 T + p T^{2}
89 1+pT2 1 + 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

−8.198352029776081704179962583997, −7.20665802211394294788507076701, −6.58224189435894862107686196920, −6.03471583126998769636769855173, −4.98010725817959605569098291710, −4.18959516479436501828774024575, −3.33632083290262564079618868160, −2.70081086208938402175006033537, −1.28374529154109134819146796964, 0, 1.28374529154109134819146796964, 2.70081086208938402175006033537, 3.33632083290262564079618868160, 4.18959516479436501828774024575, 4.98010725817959605569098291710, 6.03471583126998769636769855173, 6.58224189435894862107686196920, 7.20665802211394294788507076701, 8.198352029776081704179962583997

Graph of the ZZ-function along the critical line