Properties

Label 2-8470-1.1-c1-0-200
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.15·3-s + 4-s − 5-s + 1.15·6-s + 7-s + 8-s − 1.65·9-s − 10-s + 1.15·12-s − 3.58·13-s + 14-s − 1.15·15-s + 16-s + 4.29·17-s − 1.65·18-s − 4.89·19-s − 20-s + 1.15·21-s + 2.85·23-s + 1.15·24-s + 25-s − 3.58·26-s − 5.39·27-s + 28-s − 0.754·29-s − 1.15·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.668·3-s + 0.5·4-s − 0.447·5-s + 0.472·6-s + 0.377·7-s + 0.353·8-s − 0.552·9-s − 0.316·10-s + 0.334·12-s − 0.994·13-s + 0.267·14-s − 0.299·15-s + 0.250·16-s + 1.04·17-s − 0.391·18-s − 1.12·19-s − 0.223·20-s + 0.252·21-s + 0.595·23-s + 0.236·24-s + 0.200·25-s − 0.703·26-s − 1.03·27-s + 0.188·28-s − 0.140·29-s − 0.211·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(67.6332\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8470} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 1.15T + 3T^{2} \)
13 \( 1 + 3.58T + 13T^{2} \)
17 \( 1 - 4.29T + 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + 0.754T + 29T^{2} \)
31 \( 1 + 9.35T + 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 0.0689T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
61 \( 1 + 4.03T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 + 9.06T + 73T^{2} \)
79 \( 1 + 5.03T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 + 6.71T + 97T^{2} \)
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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

−7.39177303441412116790171712369, −6.96714724543137475980383556427, −5.77340249683022034865076333079, −5.43591755818161829370804157773, −4.50639771525269995612311012595, −3.87579090431928980764120734117, −3.08813781158215951019043592965, −2.46844559134977890564393720408, −1.56175834493196827513806965033, 0, 1.56175834493196827513806965033, 2.46844559134977890564393720408, 3.08813781158215951019043592965, 3.87579090431928980764120734117, 4.50639771525269995612311012595, 5.43591755818161829370804157773, 5.77340249683022034865076333079, 6.96714724543137475980383556427, 7.39177303441412116790171712369

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