Properties

Label 2-462-1.1-c3-0-28
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 + 5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 2·10-s − 11·11-s − 12·12-s − 43·13-s − 14·14-s − 3·15-s + 16·16-s + 100·17-s + 18·18-s − 87·19-s + 4·20-s + 21·21-s − 22·22-s − 58·23-s − 24·24-s − 124·25-s − 86·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.0894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.0632·10-s − 0.301·11-s − 0.288·12-s − 0.917·13-s − 0.267·14-s − 0.0516·15-s + 1/4·16-s + 1.42·17-s + 0.235·18-s − 1.05·19-s + 0.0447·20-s + 0.218·21-s − 0.213·22-s − 0.525·23-s − 0.204·24-s − 0.991·25-s − 0.648·26-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: $\chi_{462} (1, \cdot )$
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 \( 1 - T + p^{3} T^{2} \)
13 \( 1 + 43 T + p^{3} T^{2} \)
17 \( 1 - 100 T + p^{3} T^{2} \)
19 \( 1 + 87 T + p^{3} T^{2} \)
23 \( 1 + 58 T + p^{3} T^{2} \)
29 \( 1 + 223 T + p^{3} T^{2} \)
31 \( 1 - 88 T + p^{3} T^{2} \)
37 \( 1 - p T + p^{3} T^{2} \)
41 \( 1 - 128 T + p^{3} T^{2} \)
43 \( 1 + 458 T + p^{3} T^{2} \)
47 \( 1 + 341 T + p^{3} T^{2} \)
53 \( 1 + 342 T + p^{3} T^{2} \)
59 \( 1 + 105 T + p^{3} T^{2} \)
61 \( 1 - 190 T + p^{3} T^{2} \)
67 \( 1 + 579 T + p^{3} T^{2} \)
71 \( 1 - 128 T + p^{3} T^{2} \)
73 \( 1 + 161 T + p^{3} T^{2} \)
79 \( 1 + 396 T + p^{3} T^{2} \)
83 \( 1 + 420 T + p^{3} T^{2} \)
89 \( 1 + 798 T + p^{3} T^{2} \)
97 \( 1 - 1414 T + p^{3} T^{2} \)
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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

−10.18239039261953249571980998604, −9.678656207262640074296171864295, −8.107861945681130758424029173634, −7.26152122927396928105424834843, −6.18835380869119872570346214046, −5.46616914785977869841965449686, −4.42993825397833289775223072476, −3.26586092151655815076882289958, −1.87500502197503811572350213528, 0, 1.87500502197503811572350213528, 3.26586092151655815076882289958, 4.42993825397833289775223072476, 5.46616914785977869841965449686, 6.18835380869119872570346214046, 7.26152122927396928105424834843, 8.107861945681130758424029173634, 9.678656207262640074296171864295, 10.18239039261953249571980998604

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