Properties

Label 2-666-37.27-c1-0-2
Degree $2$
Conductor $666$
Sign $-0.0642 - 0.997i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.5 + 0.866i)5-s + (1 + 1.73i)7-s − 0.999i·8-s + 1.73·10-s + 4.73·11-s + (−3 + 1.73i)13-s − 1.99i·14-s + (−0.5 + 0.866i)16-s + (−6.69 − 3.86i)17-s + (1.09 − 0.633i)19-s + (−1.49 − 0.866i)20-s + (−4.09 − 2.36i)22-s + 4.73i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.670 + 0.387i)5-s + (0.377 + 0.654i)7-s − 0.353i·8-s + 0.547·10-s + 1.42·11-s + (−0.832 + 0.480i)13-s − 0.534i·14-s + (−0.125 + 0.216i)16-s + (−1.62 − 0.937i)17-s + (0.251 − 0.145i)19-s + (−0.335 − 0.193i)20-s + (−0.873 − 0.504i)22-s + 0.986i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0642 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0642 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.0642 - 0.997i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.0642 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500603 + 0.533844i\)
\(L(\frac12)\) \(\approx\) \(0.500603 + 0.533844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
37 \( 1 + (5.69 - 2.13i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.69 + 3.86i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.73iT - 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
41 \( 1 + (-4.96 - 8.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.928iT - 43T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.19 + 1.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 8.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 - 3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (11.4 - 6.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.83 - 4.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.89 - 3.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.73iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.99327587916121889941228599519, −9.610694136848590201991930983171, −9.150673908832041092339044792392, −8.327199107246359064407720554127, −7.10529977937107476067924107354, −6.77760474101017472573581093085, −5.16910884083183180971170721012, −4.08703494494826105389452583530, −2.93641993273734670851664719755, −1.65128066560542990496830322626, 0.48289382163635573935602160726, 2.07006677618603872908096989885, 3.98303133993218743225437657533, 4.56011928961768916119618265990, 6.04704708051329774290501367732, 6.88952748536424064428473743010, 7.75829139351221850521833489998, 8.521049846414439601263913122801, 9.258963866753378299729953283288, 10.27880790689707783747839286779

Graph of the $Z$-function along the critical line