Properties

Label 2-462-1.1-c1-0-6
Degree $2$
Conductor $462$
Sign $1$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 2·5-s + 6-s − 7-s + 8-s + 9-s + 2·10-s + 11-s + 12-s − 2·13-s − 14-s + 2·15-s + 16-s − 2·17-s + 18-s + 2·20-s − 21-s + 22-s + 24-s − 25-s − 2·26-s + 27-s − 28-s − 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.218·21-s + 0.213·22-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.757859414\)
\(L(\frac12)\) \(\approx\) \(2.757859414\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−11.07234697320449720415471226055, −10.02727748724745217892398596458, −9.432611216215457888567824586170, −8.354764684441443853355726283697, −7.16036475066804637810786253712, −6.34557563328331303195978146208, −5.33211397644666962563070735994, −4.18774926881798203669654007783, −2.96523175330119569799799482797, −1.88425382150531611231140922836, 1.88425382150531611231140922836, 2.96523175330119569799799482797, 4.18774926881798203669654007783, 5.33211397644666962563070735994, 6.34557563328331303195978146208, 7.16036475066804637810786253712, 8.354764684441443853355726283697, 9.432611216215457888567824586170, 10.02727748724745217892398596458, 11.07234697320449720415471226055

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