Properties

Label 2-8470-1.1-c1-0-111
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s + 5-s + 3·6-s − 7-s − 8-s + 6·9-s − 10-s − 3·12-s − 13-s + 14-s − 3·15-s + 16-s − 6·18-s + 7·19-s + 20-s + 3·21-s + 23-s + 3·24-s + 25-s + 26-s − 9·27-s − 28-s − 8·29-s + 3·30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.316·10-s − 0.866·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.41·18-s + 1.60·19-s + 0.223·20-s + 0.654·21-s + 0.208·23-s + 0.612·24-s + 1/5·25-s + 0.196·26-s − 1.73·27-s − 0.188·28-s − 1.48·29-s + 0.547·30-s − 0.718·31-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: yes
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 \( 1 + p T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.38364095861201726416294874423, −6.68729811721464114212437017112, −6.09468197078664885074817151272, −5.46380711015514181707768845096, −5.03033489618666878857111895897, −3.96672091259259568845386939329, −2.98905823221852683176650173332, −1.80346821273607057968830152715, −0.979308135441860368550099086647, 0, 0.979308135441860368550099086647, 1.80346821273607057968830152715, 2.98905823221852683176650173332, 3.96672091259259568845386939329, 5.03033489618666878857111895897, 5.46380711015514181707768845096, 6.09468197078664885074817151272, 6.68729811721464114212437017112, 7.38364095861201726416294874423

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