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Properties

Label 2-666-37.10-c1-0-13
Degree $2$
Conductor $666$
Sign $-0.999 - 0.0414i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−1.15 − 2.00i)7-s − 0.999·8-s − 0.999·10-s − 2.31·11-s + (−1 − 1.73i)13-s − 2.31·14-s + (−0.5 + 0.866i)16-s + (−1.65 + 2.87i)17-s + (−0.158 − 0.274i)19-s + (−0.499 + 0.866i)20-s + (−1.15 + 2.00i)22-s − 4·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.437 − 0.758i)7-s − 0.353·8-s − 0.316·10-s − 0.698·11-s + (−0.277 − 0.480i)13-s − 0.619·14-s + (−0.125 + 0.216i)16-s + (−0.402 + 0.696i)17-s + (−0.0363 − 0.0629i)19-s + (−0.111 + 0.193i)20-s + (−0.246 + 0.427i)22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.999 - 0.0414i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.999 - 0.0414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0180968 + 0.872724i\)
\(L(\frac12)\) \(\approx\) \(0.0180968 + 0.872724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
37 \( 1 + (-3.97 + 4.60i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.15 + 2.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.158 + 0.274i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 + 2.31T + 31T^{2} \)
41 \( 1 + (2.34 + 4.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.316T + 43T^{2} \)
47 \( 1 + 4.31T + 47T^{2} \)
53 \( 1 + (2.31 - 4.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.31 + 7.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.341 - 0.591i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.158 - 0.274i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.158 + 0.274i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + (3.47 + 6.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.31 + 7.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.34 - 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−10.31430629643779145520851135960, −9.421642340591497582369056144611, −8.362679924376204203912598051590, −7.52342134549982753598758735900, −6.38022474132996650544428413706, −5.34922967743570958637662526372, −4.32925456134807158443272191149, −3.46999258393496830350935504462, −2.13806388583065570767034494071, −0.39168328150696687332452967169, 2.38105576694336703510226794493, 3.42229522551384537547121727666, 4.68798763830789908906455402614, 5.59778466146541214370119250160, 6.52194098258030178554266524651, 7.34099306845876720174753007870, 8.214980468904254877638154993972, 9.187828709313595984074917357230, 9.905361637606969906756293370266, 11.11913375717252095934538151862

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