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Properties

Label 2-567-7.6-c0-0-3
Degree 22
Conductor 567567
Sign 11
Analytic cond. 0.2829690.282969
Root an. cond. 0.5319490.531949
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 1.99·4-s − 7-s + 1.73·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s − 3.46·44-s + 49-s + 1.73·50-s − 1.73·53-s − 1.73·56-s − 1.00·64-s − 67-s + 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 1.73·98-s + 1.99·100-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s − 7-s + 1.73·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 25-s − 1.99·28-s + 37-s + 43-s − 3.46·44-s + 49-s + 1.73·50-s − 1.73·53-s − 1.73·56-s − 1.00·64-s − 67-s + 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 1.73·98-s + 1.99·100-s + ⋯

Functional equation

Λ(s)=(567s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(567s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 567567    =    34⋅73^{4} \cdot 7
Sign: 11
Analytic conductor: 0.2829690.282969
Root analytic conductor: 0.5319490.531949
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ567(244,⋅)\chi_{567} (244, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 567, ( :0), 1)(2,\ 567,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9587166531.958716653
L(12)L(\frac12) ≈\approx 1.9587166531.958716653
L(1)L(1) 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)
bad3 1 1
7 1+T 1 + T
good2 1−1.73T+T2 1 - 1.73T + T^{2}
5 1−T2 1 - T^{2}
11 1+1.73T+T2 1 + 1.73T + T^{2}
13 1−T2 1 - T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1−T+T2 1 - T + T^{2}
41 1−T2 1 - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1−T2 1 - T^{2}
53 1+1.73T+T2 1 + 1.73T + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+T+T2 1 + T + T^{2}
71 1−1.73T+T2 1 - 1.73T + T^{2}
73 1−T2 1 - T^{2}
79 1+T+T2 1 + T + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - 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

−11.07067185158620858047463607831, −10.46569589200719895761701231757, −9.371246171645520668549116380100, −7.970835435007244695063973068796, −7.02666353442432348798601254878, −6.12269619030713511372617528053, −5.33569885850940032010161472943, −4.42938801666431015361494420647, −3.20497650277098718036239044932, −2.52899724031951772135360380788, 2.52899724031951772135360380788, 3.20497650277098718036239044932, 4.42938801666431015361494420647, 5.33569885850940032010161472943, 6.12269619030713511372617528053, 7.02666353442432348798601254878, 7.970835435007244695063973068796, 9.371246171645520668549116380100, 10.46569589200719895761701231757, 11.07067185158620858047463607831

Graph of the ZZ-function along the critical line