Properties

Label 2-462-1.1-c3-0-23
Degree $2$
Conductor $462$
Sign $-1$
Analytic cond. $27.2588$
Root an. cond. $5.22100$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s − 7·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 14·10-s + 11·11-s + 12·12-s − 67·13-s − 14·14-s − 21·15-s + 16·16-s + 30·17-s − 18·18-s − 7·19-s − 28·20-s + 21·21-s − 22·22-s + 28·23-s − 24·24-s − 76·25-s + 134·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.626·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.442·10-s + 0.301·11-s + 0.288·12-s − 1.42·13-s − 0.267·14-s − 0.361·15-s + 1/4·16-s + 0.428·17-s − 0.235·18-s − 0.0845·19-s − 0.313·20-s + 0.218·21-s − 0.213·22-s + 0.253·23-s − 0.204·24-s − 0.607·25-s + 1.01·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(27.2588\)
Root analytic conductor: \(5.22100\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 462,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 - p T \)
7 \( 1 - p T \)
11 \( 1 - p T \)
good5 \( 1 + 7 T + p^{3} T^{2} \)
13 \( 1 + 67 T + p^{3} T^{2} \)
17 \( 1 - 30 T + p^{3} T^{2} \)
19 \( 1 + 7 T + p^{3} T^{2} \)
23 \( 1 - 28 T + p^{3} T^{2} \)
29 \( 1 - 121 T + p^{3} T^{2} \)
31 \( 1 + 10 p T + p^{3} T^{2} \)
37 \( 1 + 71 T + p^{3} T^{2} \)
41 \( 1 + 180 T + p^{3} T^{2} \)
43 \( 1 + 108 T + p^{3} T^{2} \)
47 \( 1 - 71 T + p^{3} T^{2} \)
53 \( 1 - 128 T + p^{3} T^{2} \)
59 \( 1 + 429 T + p^{3} T^{2} \)
61 \( 1 - 22 T + p^{3} T^{2} \)
67 \( 1 + 803 T + p^{3} T^{2} \)
71 \( 1 - 468 T + p^{3} T^{2} \)
73 \( 1 + 117 T + p^{3} T^{2} \)
79 \( 1 + 96 T + p^{3} T^{2} \)
83 \( 1 + 1122 T + p^{3} T^{2} \)
89 \( 1 + 1146 T + p^{3} T^{2} \)
97 \( 1 + 92 T + p^{3} T^{2} \)
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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

−10.02093661223488596513011609690, −9.260594361098412231083034496616, −8.339840832471571152812023329613, −7.57066452820735548638508763045, −6.92161295198282181215241191329, −5.39096213503668459424458951989, −4.16601178026925040226042863566, −2.93770681074307696211309829245, −1.67182386761785152109276720802, 0, 1.67182386761785152109276720802, 2.93770681074307696211309829245, 4.16601178026925040226042863566, 5.39096213503668459424458951989, 6.92161295198282181215241191329, 7.57066452820735548638508763045, 8.339840832471571152812023329613, 9.260594361098412231083034496616, 10.02093661223488596513011609690

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