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Properties

Label 2-50575-1.1-c1-0-31
Degree 22
Conductor 5057550575
Sign −1-1
Analytic cond. 403.843403.843
Root an. cond. 20.095820.0958
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  + 2·2-s − 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s + 3·11-s − 2·12-s + 13-s + 2·14-s − 4·16-s − 4·18-s − 21-s + 6·22-s − 6·23-s + 2·26-s + 5·27-s + 2·28-s + 5·29-s − 2·31-s − 8·32-s − 3·33-s − 4·36-s − 2·37-s − 39-s − 2·41-s − 2·42-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 16-s − 0.942·18-s − 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.392·26-s + 0.962·27-s + 0.377·28-s + 0.928·29-s − 0.359·31-s − 1.41·32-s − 0.522·33-s − 2/3·36-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 0.308·42-s + ⋯

Functional equation

Λ(s)=(50575s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(50575s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5057550575    =    52⋅7⋅1725^{2} \cdot 7 \cdot 17^{2}
Sign: −1-1
Analytic conductor: 403.843403.843
Root analytic conductor: 20.095820.0958
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 50575, ( :1/2), −1)(2,\ 50575,\ (\ :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)
bad5 1 1
7 1−T 1 - T
17 1 1
good2 1−pT+pT2 1 - p T + p T^{2}
3 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+2T+pT2 1 + 2 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+3T+pT2 1 + 3 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 1−8T+pT2 1 - 8 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+7T+pT2 1 + 7 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

−14.64002564125645, −14.20978926514614, −13.81394253146794, −13.33426010054795, −12.67902887038101, −12.13225818568645, −11.80127670845848, −11.47035424197259, −10.93528669008964, −10.26341222699785, −9.693540103108074, −8.904963343182625, −8.491099265509478, −7.946341807422589, −6.913490484003947, −6.647273098803020, −6.069066533562253, −5.515894966802146, −5.144459889976130, −4.452661876647726, −3.893401299926782, −3.439176202902306, −2.616411801385783, −1.987692128314599, −1.031503631318047, 0, 1.031503631318047, 1.987692128314599, 2.616411801385783, 3.439176202902306, 3.893401299926782, 4.452661876647726, 5.144459889976130, 5.515894966802146, 6.069066533562253, 6.647273098803020, 6.913490484003947, 7.946341807422589, 8.491099265509478, 8.904963343182625, 9.693540103108074, 10.26341222699785, 10.93528669008964, 11.47035424197259, 11.80127670845848, 12.13225818568645, 12.67902887038101, 13.33426010054795, 13.81394253146794, 14.20978926514614, 14.64002564125645

Graph of the ZZ-function along the critical line