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Properties

Label 2-666-1.1-c1-0-3
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 − 2·5-s + 8-s − 2·10-s + 4·11-s + 6·13-s + 16-s − 6·17-s + 8·19-s − 2·20-s + 4·22-s − 25-s + 6·26-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 37-s + 8·38-s − 2·40-s + 6·41-s − 8·43-s + 4·44-s − 8·47-s − 7·49-s − 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 1.17·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.164·37-s + 1.29·38-s − 0.316·40-s + 0.937·41-s − 1.21·43-s + 0.603·44-s − 1.16·47-s − 49-s − 0.141·50-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 2.2246573032.224657303
L(12)L(\frac12) ≈\approx 2.2246573032.224657303
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+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 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+12T+pT2 1 + 12 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−10.89285735420782500834690215413, −9.633028066442567234094648512935, −8.674805511220596933018409082872, −7.87134380463197584693994089861, −6.73345927651178301509646905181, −6.17635572856702991761729027556, −4.79961428857637427378957471382, −3.94279778871499007659572657862, −3.16142223901297670850098979789, −1.32659579994960613230952815319, 1.32659579994960613230952815319, 3.16142223901297670850098979789, 3.94279778871499007659572657862, 4.79961428857637427378957471382, 6.17635572856702991761729027556, 6.73345927651178301509646905181, 7.87134380463197584693994089861, 8.674805511220596933018409082872, 9.633028066442567234094648512935, 10.89285735420782500834690215413

Graph of the ZZ-function along the critical line