↑

Properties

Label 2-666-1.1-c1-0-13
Degree 22
Conductor 666666
Sign −1-1
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 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·5-s − 7-s + 8-s − 4·10-s + 11-s − 3·13-s − 14-s + 16-s − 3·17-s − 5·19-s − 4·20-s + 22-s − 5·23-s + 11·25-s − 3·26-s − 28-s − 4·29-s − 10·31-s + 32-s − 3·34-s + 4·35-s − 37-s − 5·38-s − 4·40-s + 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.301·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.894·20-s + 0.213·22-s − 1.04·23-s + 11/5·25-s − 0.588·26-s − 0.188·28-s − 0.742·29-s − 1.79·31-s + 0.176·32-s − 0.514·34-s + 0.676·35-s − 0.164·37-s − 0.811·38-s − 0.632·40-s + 0.937·41-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: yes
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+4T+pT2 1 + 4 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−T+pT2 1 - 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+5T+pT2 1 + 5 T + p T^{2}
23 1+5T+pT2 1 + 5 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+10T+pT2 1 + 10 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+2T+pT2 1 + 2 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+11T+pT2 1 + 11 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{2}
show more
show less
   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

−10.34656449179909477685318487264, −9.100924645180121051837481922239, −8.162699398202549566838232688366, −7.32327856135543720727953115667, −6.66446396406084609826772404942, −5.36951137817095598841912521526, −4.13511654126083004672518336782, −3.82521452982993692459318044646, −2.37866343224007325240428184274, 0, 2.37866343224007325240428184274, 3.82521452982993692459318044646, 4.13511654126083004672518336782, 5.36951137817095598841912521526, 6.66446396406084609826772404942, 7.32327856135543720727953115667, 8.162699398202549566838232688366, 9.100924645180121051837481922239, 10.34656449179909477685318487264

Graph of the ZZ-function along the critical line