Properties

Label 2-462-231.95-c1-0-7
Degree $2$
Conductor $462$
Sign $0.713 - 0.700i$
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  + (0.104 − 0.994i)2-s + (−1.63 + 0.585i)3-s + (−0.978 − 0.207i)4-s + (0.112 − 0.251i)5-s + (0.412 + 1.68i)6-s + (−2.57 − 0.597i)7-s + (−0.309 + 0.951i)8-s + (2.31 − 1.90i)9-s + (−0.238 − 0.137i)10-s + (0.877 + 3.19i)11-s + (1.71 − 0.233i)12-s + (0.376 − 0.518i)13-s + (−0.863 + 2.50i)14-s + (−0.0352 + 0.475i)15-s + (0.913 + 0.406i)16-s + (0.638 + 6.07i)17-s + ⋯
L(s)  = 1  + (0.0739 − 0.703i)2-s + (−0.941 + 0.338i)3-s + (−0.489 − 0.103i)4-s + (0.0500 − 0.112i)5-s + (0.168 + 0.686i)6-s + (−0.974 − 0.225i)7-s + (−0.109 + 0.336i)8-s + (0.771 − 0.636i)9-s + (−0.0754 − 0.0435i)10-s + (0.264 + 0.964i)11-s + (0.495 − 0.0675i)12-s + (0.104 − 0.143i)13-s + (−0.230 + 0.668i)14-s + (−0.00910 + 0.122i)15-s + (0.228 + 0.101i)16-s + (0.154 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674370 + 0.275487i\)
\(L(\frac12)\) \(\approx\) \(0.674370 + 0.275487i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (1.63 - 0.585i)T \)
7 \( 1 + (2.57 + 0.597i)T \)
11 \( 1 + (-0.877 - 3.19i)T \)
good5 \( 1 + (-0.112 + 0.251i)T + (-3.34 - 3.71i)T^{2} \)
13 \( 1 + (-0.376 + 0.518i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.638 - 6.07i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-0.228 - 1.07i)T + (-17.3 + 7.72i)T^{2} \)
23 \( 1 + (-1.05 + 0.608i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.73 - 8.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.57 - 2.03i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-4.83 + 5.37i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (0.939 - 2.89i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.12iT - 43T^{2} \)
47 \( 1 + (0.496 + 2.33i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (3.33 + 7.48i)T + (-35.4 + 39.3i)T^{2} \)
59 \( 1 + (1.83 - 8.65i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (1.80 - 4.04i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (5.31 - 9.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.62 + 9.11i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.37 + 11.1i)T + (-66.6 - 29.6i)T^{2} \)
79 \( 1 + (-12.3 - 1.29i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (3.41 - 2.48i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (15.7 - 9.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.14 + 5.91i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.94070629706199170787130340072, −10.44852145003585762956989451628, −9.648735130320275986707096072523, −8.854305931198840098532638487842, −7.28772048514412287189257139067, −6.37578361796616445707351685870, −5.36719129610794633652447062719, −4.27171772685615324893824262016, −3.34425018309058149940573488302, −1.42751269587881556672406063268, 0.53949717027603097291103729771, 2.90675575697275070018296307585, 4.40151627514292613711414225526, 5.54359985075755950400205036032, 6.30141069926169073224956903706, 6.95626577096031331700397380025, 7.999760076426116916043697664363, 9.197088106042818832258788010397, 9.929105218933532102288134200364, 11.08218530598228970345864841588

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