Properties

Label 2-8470-1.1-c1-0-179
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 − 0.247·3-s + 4-s − 5-s − 0.247·6-s + 7-s + 8-s − 2.93·9-s − 10-s − 0.247·12-s + 2.83·13-s + 14-s + 0.247·15-s + 16-s − 5.76·17-s − 2.93·18-s + 3.66·19-s − 20-s − 0.247·21-s − 7.17·23-s − 0.247·24-s + 25-s + 2.83·26-s + 1.46·27-s + 28-s + 1.41·29-s + 0.247·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.142·3-s + 0.5·4-s − 0.447·5-s − 0.100·6-s + 0.377·7-s + 0.353·8-s − 0.979·9-s − 0.316·10-s − 0.0713·12-s + 0.786·13-s + 0.267·14-s + 0.0638·15-s + 0.250·16-s − 1.39·17-s − 0.692·18-s + 0.841·19-s − 0.223·20-s − 0.0539·21-s − 1.49·23-s − 0.0504·24-s + 0.200·25-s + 0.555·26-s + 0.282·27-s + 0.188·28-s + 0.262·29-s + 0.0451·30-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 \( 1 + 0.247T + 3T^{2} \)
13 \( 1 - 2.83T + 13T^{2} \)
17 \( 1 + 5.76T + 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + 7.17T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 0.356T + 31T^{2} \)
37 \( 1 - 2.94T + 37T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 9.64T + 43T^{2} \)
47 \( 1 - 3.13T + 47T^{2} \)
53 \( 1 - 0.874T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 7.19T + 73T^{2} \)
79 \( 1 - 6.14T + 79T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.57960859204042089983702180262, −6.49349798756401979141820987102, −6.03182576322721507744799240262, −5.42769996125539195541720060624, −4.45888000436320847807961721725, −4.08737443061898819103000287647, −3.09945970804122573884860295251, −2.43608495234153353703544069169, −1.35841165360969547260480573950, 0, 1.35841165360969547260480573950, 2.43608495234153353703544069169, 3.09945970804122573884860295251, 4.08737443061898819103000287647, 4.45888000436320847807961721725, 5.42769996125539195541720060624, 6.03182576322721507744799240262, 6.49349798756401979141820987102, 7.57960859204042089983702180262

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