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Properties

Label 2-966-1.1-c1-0-17
Degree 22
Conductor 966966
Sign 11
Analytic cond. 7.713547.71354
Root an. cond. 2.777322.77732
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-s + 3-s + 4-s + 3·5-s + 6-s − 7-s + 8-s + 9-s + 3·10-s + 4·11-s + 12-s − 3·13-s − 14-s + 3·15-s + 16-s − 4·17-s + 18-s + 3·20-s − 21-s + 4·22-s + 23-s + 24-s + 4·25-s − 3·26-s + 27-s − 28-s + 3·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.670·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯

Functional equation

Λ(s)=(966s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(966s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 966966    =    2⋅3⋅7⋅232 \cdot 3 \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 7.713547.71354
Root analytic conductor: 2.777322.77732
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 966, ( :1/2), 1)(2,\ 966,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5935270083.593527008
L(12)L(\frac12) ≈\approx 3.5935270083.593527008
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−T 1 - T
3 1−T 1 - T
7 1+T 1 + T
23 1−T 1 - T
good5 1−3T+pT2 1 - 3 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+6T+pT2 1 + 6 T + p T^{2}
37 1+9T+pT2 1 + 9 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+3T+pT2 1 + 3 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+14T+pT2 1 + 14 T + 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

−9.843827071515005252615180187856, −9.345059082219016767573214421285, −8.533329024037475026270064025246, −7.11624966976560567011680890410, −6.61433587019532483612650095921, −5.70303300160941996822497347087, −4.72846979566270558944303333386, −3.67780852772662808140519843755, −2.54506917140467324126940654920, −1.68100316040744687359155389371, 1.68100316040744687359155389371, 2.54506917140467324126940654920, 3.67780852772662808140519843755, 4.72846979566270558944303333386, 5.70303300160941996822497347087, 6.61433587019532483612650095921, 7.11624966976560567011680890410, 8.533329024037475026270064025246, 9.345059082219016767573214421285, 9.843827071515005252615180187856

Graph of the ZZ-function along the critical line