Properties

Label 2-8470-1.1-c1-0-116
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 − 2.41·3-s + 4-s + 5-s + 2.41·6-s + 7-s − 8-s + 2.82·9-s − 10-s − 2.41·12-s − 2.76·13-s − 14-s − 2.41·15-s + 16-s − 4.44·17-s − 2.82·18-s + 2.62·19-s + 20-s − 2.41·21-s + 1.54·23-s + 2.41·24-s + 25-s + 2.76·26-s + 0.414·27-s + 28-s + 9.32·29-s + 2.41·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.447·5-s + 0.985·6-s + 0.377·7-s − 0.353·8-s + 0.942·9-s − 0.316·10-s − 0.696·12-s − 0.767·13-s − 0.267·14-s − 0.623·15-s + 0.250·16-s − 1.07·17-s − 0.666·18-s + 0.601·19-s + 0.223·20-s − 0.526·21-s + 0.322·23-s + 0.492·24-s + 0.200·25-s + 0.542·26-s + 0.0797·27-s + 0.188·28-s + 1.73·29-s + 0.440·30-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 \( 1 + 2.41T + 3T^{2} \)
13 \( 1 + 2.76T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 - 2.62T + 19T^{2} \)
23 \( 1 - 1.54T + 23T^{2} \)
29 \( 1 - 9.32T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 4.39T + 37T^{2} \)
41 \( 1 + 4.80T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 9.77T + 53T^{2} \)
59 \( 1 - 0.510T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 3.41T + 73T^{2} \)
79 \( 1 + 2.04T + 79T^{2} \)
83 \( 1 + 4.27T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 2.35T + 97T^{2} \)
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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

−7.19651559691758428097504759716, −6.81436160709463023798764602477, −6.17417830703529147587316077754, −5.24760676318334711282216852906, −5.07615203157050540339042062051, −4.04735172450996448913118132339, −2.81975094012488672893216423052, −1.96033180228123892551143595526, −0.998479830940473430376323975256, 0, 0.998479830940473430376323975256, 1.96033180228123892551143595526, 2.81975094012488672893216423052, 4.04735172450996448913118132339, 5.07615203157050540339042062051, 5.24760676318334711282216852906, 6.17417830703529147587316077754, 6.81436160709463023798764602477, 7.19651559691758428097504759716

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