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Properties

Label 2-666-1.1-c1-0-9
Degree 22
Conductor 666666
Sign −1-1
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 1.56·7-s − 8-s + 2·10-s + 1.56·11-s + 6.68·13-s + 1.56·14-s + 16-s − 3.56·17-s − 3.56·19-s − 2·20-s − 1.56·22-s − 8.68·23-s − 25-s − 6.68·26-s − 1.56·28-s + 1.12·29-s − 9.12·31-s − 32-s + 3.56·34-s + 3.12·35-s − 37-s + 3.56·38-s + 2·40-s − 11.1·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.590·7-s − 0.353·8-s + 0.632·10-s + 0.470·11-s + 1.85·13-s + 0.417·14-s + 0.250·16-s − 0.863·17-s − 0.817·19-s − 0.447·20-s − 0.332·22-s − 1.81·23-s − 0.200·25-s − 1.31·26-s − 0.295·28-s + 0.208·29-s − 1.63·31-s − 0.176·32-s + 0.610·34-s + 0.527·35-s − 0.164·37-s + 0.577·38-s + 0.316·40-s − 1.73·41-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: −1-1
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 666, ( :1/2), −1)(2,\ 666,\ (\ :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)
bad2 1+T 1 + T
3 1 1
37 1+T 1 + T
good5 1+2T+5T2 1 + 2T + 5T^{2}
7 1+1.56T+7T2 1 + 1.56T + 7T^{2}
11 1−1.56T+11T2 1 - 1.56T + 11T^{2}
13 1−6.68T+13T2 1 - 6.68T + 13T^{2}
17 1+3.56T+17T2 1 + 3.56T + 17T^{2}
19 1+3.56T+19T2 1 + 3.56T + 19T^{2}
23 1+8.68T+23T2 1 + 8.68T + 23T^{2}
29 1−1.12T+29T2 1 - 1.12T + 29T^{2}
31 1+9.12T+31T2 1 + 9.12T + 31T^{2}
41 1+11.1T+41T2 1 + 11.1T + 41T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1−10.2T+47T2 1 - 10.2T + 47T^{2}
53 1−1.56T+53T2 1 - 1.56T + 53T^{2}
59 1+0.876T+59T2 1 + 0.876T + 59T^{2}
61 1+12.2T+61T2 1 + 12.2T + 61T^{2}
67 1−2.24T+67T2 1 - 2.24T + 67T^{2}
71 1+2.24T+71T2 1 + 2.24T + 71T^{2}
73 1−3.56T+73T2 1 - 3.56T + 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1+14.9T+83T2 1 + 14.9T + 83T^{2}
89 1−12.9T+89T2 1 - 12.9T + 89T^{2}
97 1−2.87T+97T2 1 - 2.87T + 97T^{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

−10.13584358345770401042953234905, −8.975123713168213773965377021203, −8.523746805209980316711267621929, −7.59496725573606989190371983421, −6.55607500307732686649523299053, −5.93492862049092353964252060622, −4.13297606196451654133338922812, −3.51422167165774756982684200647, −1.81225175702583064467541270675, 0, 1.81225175702583064467541270675, 3.51422167165774756982684200647, 4.13297606196451654133338922812, 5.93492862049092353964252060622, 6.55607500307732686649523299053, 7.59496725573606989190371983421, 8.523746805209980316711267621929, 8.975123713168213773965377021203, 10.13584358345770401042953234905

Graph of the ZZ-function along the critical line