Properties

Label 2-8470-1.1-c1-0-1
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 $0$

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: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1094645667\)
\(L(\frac12)\) \(\approx\) \(0.1094645667\)
\(L(\frac{3}{2})\) 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

−7.47457357316189634376253987172, −7.20744122059375728177483028332, −6.48692485350260904008595465539, −5.69247214297519706222268382586, −5.25422441992433816362159916240, −4.42786465778952395258857721513, −3.58001718870563191857386375534, −2.43078767154967077120453590913, −1.38835363507498415084862762288, −0.20629108514192857871444970373, 0.20629108514192857871444970373, 1.38835363507498415084862762288, 2.43078767154967077120453590913, 3.58001718870563191857386375534, 4.42786465778952395258857721513, 5.25422441992433816362159916240, 5.69247214297519706222268382586, 6.48692485350260904008595465539, 7.20744122059375728177483028332, 7.47457357316189634376253987172

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