Properties

Label 2-8470-1.1-c1-0-149
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 − 2.95·3-s + 4-s − 5-s − 2.95·6-s + 7-s + 8-s + 5.71·9-s − 10-s − 2.95·12-s + 5.80·13-s + 14-s + 2.95·15-s + 16-s − 0.469·17-s + 5.71·18-s − 4.04·19-s − 20-s − 2.95·21-s − 7.69·23-s − 2.95·24-s + 25-s + 5.80·26-s − 8.00·27-s + 28-s + 2.63·29-s + 2.95·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.70·3-s + 0.5·4-s − 0.447·5-s − 1.20·6-s + 0.377·7-s + 0.353·8-s + 1.90·9-s − 0.316·10-s − 0.852·12-s + 1.60·13-s + 0.267·14-s + 0.762·15-s + 0.250·16-s − 0.113·17-s + 1.34·18-s − 0.928·19-s − 0.223·20-s − 0.644·21-s − 1.60·23-s − 0.602·24-s + 0.200·25-s + 1.13·26-s − 1.54·27-s + 0.188·28-s + 0.488·29-s + 0.538·30-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 \( 1 + 2.95T + 3T^{2} \)
13 \( 1 - 5.80T + 13T^{2} \)
17 \( 1 + 0.469T + 17T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 - 2.63T + 29T^{2} \)
31 \( 1 + 3.22T + 31T^{2} \)
37 \( 1 + 4.56T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
43 \( 1 + 0.790T + 43T^{2} \)
47 \( 1 + 4.28T + 47T^{2} \)
53 \( 1 - 13.9T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 6.69T + 61T^{2} \)
67 \( 1 + 6.73T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 3.95T + 79T^{2} \)
83 \( 1 + 9.44T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−7.08419850544799107467397852415, −6.53771657260069842374633438365, −5.91877879399534447124142931040, −5.54088013190963640715571558342, −4.58467318685541233387182275856, −4.16509218884189341552647559557, −3.45154993010908848864252007163, −2.02786599958810357895669064796, −1.18191539225703612469118377861, 0, 1.18191539225703612469118377861, 2.02786599958810357895669064796, 3.45154993010908848864252007163, 4.16509218884189341552647559557, 4.58467318685541233387182275856, 5.54088013190963640715571558342, 5.91877879399534447124142931040, 6.53771657260069842374633438365, 7.08419850544799107467397852415

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