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Properties

Label 2-3920-1.1-c1-0-31
Degree 22
Conductor 39203920
Sign 11
Analytic cond. 31.301331.3013
Root an. cond. 5.594765.59476
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  + 2·3-s − 5-s + 9-s − 3·11-s + 5·13-s − 2·15-s + 6·17-s + 19-s − 3·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 6·33-s + 11·37-s + 10·39-s + 3·41-s + 10·43-s − 45-s − 3·47-s + 12·51-s + 3·53-s + 3·55-s + 2·57-s − 4·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s − 0.625·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s + 1.80·37-s + 1.60·39-s + 0.468·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 1.68·51-s + 0.412·53-s + 0.404·55-s + 0.264·57-s − 0.512·61-s − 0.620·65-s + 0.488·67-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 31.301331.3013
Root analytic conductor: 5.594765.59476
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3920, ( :1/2), 1)(2,\ 3920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7275875342.727587534
L(12)L(\frac12) ≈\approx 2.7275875342.727587534
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
7 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−T+pT2 1 - 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 1−4T+pT2 1 - 4 T + p T^{2}
37 1−11T+pT2 1 - 11 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.312353373911425463378423277597, −7.81189036994320081381812287447, −7.47025394478870762896015822967, −6.09431538082891075849115819441, −5.65372048903059002115277189268, −4.45082680945347931104217752177, −3.63096321073868899269924714815, −3.08375678221313543410500442005, −2.17787868933866687584737323067, −0.919024052519317686191599815600, 0.919024052519317686191599815600, 2.17787868933866687584737323067, 3.08375678221313543410500442005, 3.63096321073868899269924714815, 4.45082680945347931104217752177, 5.65372048903059002115277189268, 6.09431538082891075849115819441, 7.47025394478870762896015822967, 7.81189036994320081381812287447, 8.312353373911425463378423277597

Graph of the ZZ-function along the critical line