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Properties

Label 2-462-33.8-c1-0-21
Degree $2$
Conductor $462$
Sign $0.457 + 0.889i$
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  + (0.809 − 0.587i)2-s + (1.65 − 0.500i)3-s + (0.309 − 0.951i)4-s + (0.732 − 1.00i)5-s + (1.04 − 1.37i)6-s + (0.951 + 0.309i)7-s + (−0.309 − 0.951i)8-s + (2.49 − 1.66i)9-s − 1.24i·10-s + (−3.01 + 1.37i)11-s + (0.0358 − 1.73i)12-s + (1.55 + 2.14i)13-s + (0.951 − 0.309i)14-s + (0.709 − 2.03i)15-s + (−0.809 − 0.587i)16-s + (−0.0657 − 0.0477i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.957 − 0.289i)3-s + (0.154 − 0.475i)4-s + (0.327 − 0.450i)5-s + (0.427 − 0.563i)6-s + (0.359 + 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.832 − 0.553i)9-s − 0.394i·10-s + (−0.910 + 0.413i)11-s + (0.0103 − 0.499i)12-s + (0.431 + 0.594i)13-s + (0.254 − 0.0825i)14-s + (0.183 − 0.526i)15-s + (−0.202 − 0.146i)16-s + (−0.0159 − 0.0115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.457 + 0.889i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.457 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28973 - 1.39699i\)
\(L(\frac12)\) \(\approx\) \(2.28973 - 1.39699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-1.65 + 0.500i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (3.01 - 1.37i)T \)
good5 \( 1 + (-0.732 + 1.00i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.55 - 2.14i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0657 + 0.0477i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.03 - 1.96i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.41iT - 23T^{2} \)
29 \( 1 + (0.273 - 0.843i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.202 - 0.147i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.01 - 3.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.35 + 4.18i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.86iT - 43T^{2} \)
47 \( 1 + (-6.48 + 2.10i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.70 + 2.33i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.184 - 0.0600i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.87 - 6.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.46T + 67T^{2} \)
71 \( 1 + (7.04 - 9.69i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-15.7 - 5.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.39 + 7.42i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.69 - 2.68i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-10.4 + 7.61i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.82414247568139197239413135502, −10.07904892022318608020155667588, −8.995470935059514363496659583828, −8.362446052805868363896607658291, −7.22877024154456166583791981279, −6.15935771658553098977460194490, −4.89589380020185556080280533549, −3.99693330787179443277425861716, −2.60263824310779322832357243031, −1.64058896519311247793995917812, 2.24421400161890133044715496831, 3.24898176448525847717168277236, 4.37402740126900009685167115322, 5.46366379604634187219204315562, 6.56898761657841149082367952167, 7.69230064046442395877283408630, 8.292285495011218071889520966129, 9.259833299514811660644123597803, 10.53318952070096410621394365944, 10.87127451493954963598559055593

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