Properties

Label 2-8470-1.1-c1-0-101
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 − 0.919·3-s + 4-s − 5-s + 0.919·6-s − 7-s − 8-s − 2.15·9-s + 10-s − 0.919·12-s + 3.72·13-s + 14-s + 0.919·15-s + 16-s − 1.31·17-s + 2.15·18-s − 2.70·19-s − 20-s + 0.919·21-s − 2·23-s + 0.919·24-s + 25-s − 3.72·26-s + 4.73·27-s − 28-s − 2.95·29-s − 0.919·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.530·3-s + 0.5·4-s − 0.447·5-s + 0.375·6-s − 0.377·7-s − 0.353·8-s − 0.718·9-s + 0.316·10-s − 0.265·12-s + 1.03·13-s + 0.267·14-s + 0.237·15-s + 0.250·16-s − 0.319·17-s + 0.507·18-s − 0.621·19-s − 0.223·20-s + 0.200·21-s − 0.417·23-s + 0.187·24-s + 0.200·25-s − 0.730·26-s + 0.911·27-s − 0.188·28-s − 0.547·29-s − 0.167·30-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 \( 1 + 0.919T + 3T^{2} \)
13 \( 1 - 3.72T + 13T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 8.04T + 37T^{2} \)
41 \( 1 - 3.24T + 41T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 2.83T + 59T^{2} \)
61 \( 1 - 1.44T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 - 8.03T + 71T^{2} \)
73 \( 1 + 1.33T + 73T^{2} \)
79 \( 1 + 7.72T + 79T^{2} \)
83 \( 1 - 0.285T + 83T^{2} \)
89 \( 1 + 8.14T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−7.47039775710737469499295843865, −6.71509800933288726745357503110, −6.16227794152619864623087455557, −5.62897212247227784101093436836, −4.64019516583313422808916042586, −3.79291408372660069324239826330, −3.03434879977726604559962535629, −2.11410105784859999534708810100, −0.938450096941583789094546843435, 0, 0.938450096941583789094546843435, 2.11410105784859999534708810100, 3.03434879977726604559962535629, 3.79291408372660069324239826330, 4.64019516583313422808916042586, 5.62897212247227784101093436836, 6.16227794152619864623087455557, 6.71509800933288726745357503110, 7.47039775710737469499295843865

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