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Properties

Label 2-666-1.1-c1-0-10
Degree 22
Conductor 666666
Sign 11
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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 + 4-s + 4·5-s + 3·7-s + 8-s + 4·10-s − 5·11-s + 3·13-s + 3·14-s + 16-s − 3·17-s − 7·19-s + 4·20-s − 5·22-s − 9·23-s + 11·25-s + 3·26-s + 3·28-s − 2·31-s + 32-s − 3·34-s + 12·35-s + 37-s − 7·38-s + 4·40-s − 6·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.13·7-s + 0.353·8-s + 1.26·10-s − 1.50·11-s + 0.832·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.60·19-s + 0.894·20-s − 1.06·22-s − 1.87·23-s + 11/5·25-s + 0.588·26-s + 0.566·28-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 2.02·35-s + 0.164·37-s − 1.13·38-s + 0.632·40-s − 0.937·41-s + 0.609·43-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 11
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 1)(2,\ 666,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0680287443.068028744
L(12)L(\frac12) ≈\approx 3.0680287443.068028744
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 1
37 1−T 1 - T
good5 1−4T+pT2 1 - 4 T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−3T+pT2 1 - 3 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1+9T+pT2 1 + 9 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−6T+pT2 1 - 6 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−13T+pT2 1 - 13 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+5T+pT2 1 + 5 T + p T^{2}
89 1+11T+pT2 1 + 11 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−10.66123771129793013895319012928, −9.914305521533027310416702610710, −8.668480330778590791299654356016, −7.989549022432806984304290222280, −6.61856061275031960976062588393, −5.83968165780663709630491739946, −5.19848296413414906626093432671, −4.18804468941342822592597020971, −2.40265859153889095661502631676, −1.88395043108710714192663999988, 1.88395043108710714192663999988, 2.40265859153889095661502631676, 4.18804468941342822592597020971, 5.19848296413414906626093432671, 5.83968165780663709630491739946, 6.61856061275031960976062588393, 7.989549022432806984304290222280, 8.668480330778590791299654356016, 9.914305521533027310416702610710, 10.66123771129793013895319012928

Graph of the ZZ-function along the critical line