Properties

Label 2-464-1.1-c3-0-39
Degree $2$
Conductor $464$
Sign $-1$
Analytic cond. $27.3768$
Root an. cond. $5.23229$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.46·3-s + 2.14·5-s − 20.3·7-s + 14.7·9-s − 52.0·11-s + 7.04·13-s + 13.8·15-s + 28.7·17-s − 76.4·19-s − 131.·21-s − 59.7·23-s − 120.·25-s − 79.0·27-s − 29·29-s + 3.25·31-s − 336.·33-s − 43.6·35-s + 150.·37-s + 45.5·39-s − 92.3·41-s + 100.·43-s + 31.6·45-s − 324.·47-s + 71.4·49-s + 185.·51-s + 374.·53-s − 111.·55-s + ⋯
L(s)  = 1  + 1.24·3-s + 0.191·5-s − 1.09·7-s + 0.547·9-s − 1.42·11-s + 0.150·13-s + 0.238·15-s + 0.410·17-s − 0.922·19-s − 1.36·21-s − 0.541·23-s − 0.963·25-s − 0.563·27-s − 0.185·29-s + 0.0188·31-s − 1.77·33-s − 0.210·35-s + 0.669·37-s + 0.186·39-s − 0.351·41-s + 0.357·43-s + 0.104·45-s − 1.00·47-s + 0.208·49-s + 0.510·51-s + 0.971·53-s − 0.273·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-1$
Analytic conductor: \(27.3768\)
Root analytic conductor: \(5.23229\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 464,\ (\ :3/2),\ -1)\)

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 \( 1 \)
29 \( 1 + 29T \)
good3 \( 1 - 6.46T + 27T^{2} \)
5 \( 1 - 2.14T + 125T^{2} \)
7 \( 1 + 20.3T + 343T^{2} \)
11 \( 1 + 52.0T + 1.33e3T^{2} \)
13 \( 1 - 7.04T + 2.19e3T^{2} \)
17 \( 1 - 28.7T + 4.91e3T^{2} \)
19 \( 1 + 76.4T + 6.85e3T^{2} \)
23 \( 1 + 59.7T + 1.21e4T^{2} \)
31 \( 1 - 3.25T + 2.97e4T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + 92.3T + 6.89e4T^{2} \)
43 \( 1 - 100.T + 7.95e4T^{2} \)
47 \( 1 + 324.T + 1.03e5T^{2} \)
53 \( 1 - 374.T + 1.48e5T^{2} \)
59 \( 1 + 489.T + 2.05e5T^{2} \)
61 \( 1 - 221.T + 2.26e5T^{2} \)
67 \( 1 - 427.T + 3.00e5T^{2} \)
71 \( 1 - 898.T + 3.57e5T^{2} \)
73 \( 1 + 1.08e3T + 3.89e5T^{2} \)
79 \( 1 - 798.T + 4.93e5T^{2} \)
83 \( 1 - 436.T + 5.71e5T^{2} \)
89 \( 1 - 456.T + 7.04e5T^{2} \)
97 \( 1 - 803.T + 9.12e5T^{2} \)
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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

−9.945579417936952772682051629008, −9.359614759373141012643603027815, −8.274632773585695805230944271142, −7.73857602003709735345944209640, −6.48964043109466471883188891114, −5.48269364170251366001446266873, −3.96511131067013773180255153267, −2.99756241633376496803441875665, −2.14981094286494303634596108055, 0, 2.14981094286494303634596108055, 2.99756241633376496803441875665, 3.96511131067013773180255153267, 5.48269364170251366001446266873, 6.48964043109466471883188891114, 7.73857602003709735345944209640, 8.274632773585695805230944271142, 9.359614759373141012643603027815, 9.945579417936952772682051629008

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