Properties

Label 2-462-1.1-c1-0-3
Degree $2$
Conductor $462$
Sign $1$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 3.46·5-s − 6-s + 7-s − 8-s + 9-s − 3.46·10-s + 11-s + 12-s + 2·13-s − 14-s + 3.46·15-s + 16-s − 3.46·17-s − 18-s − 1.46·19-s + 3.46·20-s + 21-s − 22-s − 6.92·23-s − 24-s + 6.99·25-s − 2·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.54·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.894·15-s + 0.250·16-s − 0.840·17-s − 0.235·18-s − 0.335·19-s + 0.774·20-s + 0.218·21-s − 0.213·22-s − 1.44·23-s − 0.204·24-s + 1.39·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.643760195\)
\(L(\frac12)\) \(\approx\) \(1.643760195\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 - 3.46T + 5T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.74433393642895475367766421449, −10.00110105167259080413415084245, −9.214714422376758870179865567047, −8.617718601682383826649694374982, −7.53358436179697876158945181235, −6.39419680149113395802921236634, −5.67761341162773234140283236806, −4.11932092109000370232963063510, −2.46076100486846570220703570998, −1.62497531587841803918805722140, 1.62497531587841803918805722140, 2.46076100486846570220703570998, 4.11932092109000370232963063510, 5.67761341162773234140283236806, 6.39419680149113395802921236634, 7.53358436179697876158945181235, 8.617718601682383826649694374982, 9.214714422376758870179865567047, 10.00110105167259080413415084245, 10.74433393642895475367766421449

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