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Properties

Label 2-567-7.6-c0-0-0
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 0.41599673360.4159967336
L(12)L(\frac12) ≈\approx 0.41599673360.4159967336
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

−10.65655710474212145386142787143, −9.869837757445742534581493883818, −9.135096920506841633084177805828, −8.712448766472471633899433084785, −7.43575546979320228008489387266, −6.74613410204020087286201809739, −6.01180283237861809057355223694, −4.09182102344938789480164144096, −2.73761425256176880608674184749, −1.17964753789132369652410274054, 1.17964753789132369652410274054, 2.73761425256176880608674184749, 4.09182102344938789480164144096, 6.01180283237861809057355223694, 6.74613410204020087286201809739, 7.43575546979320228008489387266, 8.712448766472471633899433084785, 9.135096920506841633084177805828, 9.869837757445742534581493883818, 10.65655710474212145386142787143

Graph of the ZZ-function along the critical line