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Properties

Label 2-5760-1.1-c1-0-20
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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 − 2·11-s + 6·13-s + 6·17-s − 6·19-s − 2·23-s + 25-s − 2·29-s − 4·31-s − 2·35-s + 10·37-s + 2·41-s + 8·43-s + 6·47-s − 3·49-s − 6·53-s − 2·55-s + 10·59-s − 14·61-s + 6·65-s − 8·67-s + 8·71-s + 2·73-s + 4·77-s + 12·83-s + 6·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.603·11-s + 1.66·13-s + 1.45·17-s − 1.37·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 0.269·55-s + 1.30·59-s − 1.79·61-s + 0.744·65-s − 0.977·67-s + 0.949·71-s + 0.234·73-s + 0.455·77-s + 1.31·83-s + 0.650·85-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9805719621.980571962
L(12)L(\frac12) ≈\approx 1.9805719621.980571962
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
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−8.048710042768485330930516395035, −7.54274612857108858820721388934, −6.42080235754977654824987474304, −6.04719110162633083298590434640, −5.49136016402297728302609681539, −4.33855005895802000696144305436, −3.63038835248572951269925071912, −2.86080448090208125395391738294, −1.86532930924500865714100417741, −0.75274446480860796113194564849, 0.75274446480860796113194564849, 1.86532930924500865714100417741, 2.86080448090208125395391738294, 3.63038835248572951269925071912, 4.33855005895802000696144305436, 5.49136016402297728302609681539, 6.04719110162633083298590434640, 6.42080235754977654824987474304, 7.54274612857108858820721388934, 8.048710042768485330930516395035

Graph of the ZZ-function along the critical line