Properties

Label 2-462-77.54-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.123 - 0.992i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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 + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (1.54 + 0.893i)5-s + 0.999·6-s + (−0.165 + 2.64i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.893 − 1.54i)10-s + (2.46 + 2.22i)11-s + (−0.866 − 0.499i)12-s − 6.37·13-s + (1.46 − 2.20i)14-s − 1.78·15-s + (−0.5 + 0.866i)16-s + (−0.0530 − 0.0918i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.692 + 0.399i)5-s + 0.408·6-s + (−0.0625 + 0.998i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.282 − 0.489i)10-s + (0.742 + 0.669i)11-s + (−0.249 − 0.144i)12-s − 1.76·13-s + (0.391 − 0.589i)14-s − 0.461·15-s + (−0.125 + 0.216i)16-s + (−0.0128 − 0.0222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.123 - 0.992i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.123 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516538 + 0.584911i\)
\(L(\frac12)\) \(\approx\) \(0.516538 + 0.584911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.165 - 2.64i)T \)
11 \( 1 + (-2.46 - 2.22i)T \)
good5 \( 1 + (-1.54 - 0.893i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 6.37T + 13T^{2} \)
17 \( 1 + (0.0530 + 0.0918i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.07 + 3.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.97 - 6.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.65iT - 29T^{2} \)
31 \( 1 + (-1.10 + 0.639i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.26 - 3.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.0321T + 41T^{2} \)
43 \( 1 - 6.87iT - 43T^{2} \)
47 \( 1 + (-8.55 - 4.94i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.313 - 0.543i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.7 + 6.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.97 - 8.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.42T + 71T^{2} \)
73 \( 1 + (0.0625 + 0.108i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.80 - 5.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (4.64 + 2.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−11.32666108045729560641175001327, −10.12386337822868179852407790261, −9.635675648007245817577274744825, −9.027399670050351180539362486639, −7.57149620822893923946259561012, −6.73780893517968632484320476576, −5.66268700303407675347747429732, −4.65869088695680962131702964749, −2.97963833672219119071455242424, −1.88032023741707717066010688505, 0.61236163220150357328390656309, 2.12286420418478248789033528172, 4.11522890977416855874149102723, 5.34046522460985630084996708975, 6.23320233274011371475948133610, 7.15087197727363438993948003798, 7.942673922404105006048915020409, 9.102637339085609143768443495419, 10.02004334558004644904667964876, 10.45111280815620937128081652795

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