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Properties

Label 2-640-1.1-c1-0-14
Degree 22
Conductor 640640
Sign −1-1
Analytic cond. 5.110425.11042
Root an. cond. 2.260622.26062
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 − 2·7-s − 3·9-s − 6·11-s + 2·13-s − 6·17-s + 2·19-s − 6·23-s + 25-s + 6·29-s − 4·31-s − 2·35-s + 6·37-s − 2·41-s − 4·43-s − 3·45-s + 10·47-s − 3·49-s + 2·53-s − 6·55-s − 10·59-s − 10·61-s + 6·63-s + 2·65-s + 4·67-s − 16·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 9-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.45·47-s − 3/7·49-s + 0.274·53-s − 0.809·55-s − 1.30·59-s − 1.28·61-s + 0.755·63-s + 0.248·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s + ⋯

Functional equation

Λ(s)=(640s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: −1-1
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 640, ( :1/2), −1)(2,\ 640,\ (\ :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
5 1−T 1 - T
good3 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 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 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−6T+pT2 1 - 6 T + 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

−10.27119522394318848101695814883, −9.240805137182251408961722561285, −8.459530406419082967845984524867, −7.57454589175301650317308735633, −6.31867615323075833576541028992, −5.74481417491481656913857127073, −4.64601077885432605050562198244, −3.15828578541213785461358104860, −2.31650484656215274723281292429, 0, 2.31650484656215274723281292429, 3.15828578541213785461358104860, 4.64601077885432605050562198244, 5.74481417491481656913857127073, 6.31867615323075833576541028992, 7.57454589175301650317308735633, 8.459530406419082967845984524867, 9.240805137182251408961722561285, 10.27119522394318848101695814883

Graph of the ZZ-function along the critical line