Properties

Label 2-8470-1.1-c1-0-197
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 + 2.82·3-s + 4-s − 5-s − 2.82·6-s + 7-s − 8-s + 5.00·9-s + 10-s + 2.82·12-s − 2·13-s − 14-s − 2.82·15-s + 16-s − 7.65·17-s − 5.00·18-s + 2.82·19-s − 20-s + 2.82·21-s − 2.82·23-s − 2.82·24-s + 25-s + 2·26-s + 5.65·27-s + 28-s − 3.17·29-s + 2.82·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.447·5-s − 1.15·6-s + 0.377·7-s − 0.353·8-s + 1.66·9-s + 0.316·10-s + 0.816·12-s − 0.554·13-s − 0.267·14-s − 0.730·15-s + 0.250·16-s − 1.85·17-s − 1.17·18-s + 0.648·19-s − 0.223·20-s + 0.617·21-s − 0.589·23-s − 0.577·24-s + 0.200·25-s + 0.392·26-s + 1.08·27-s + 0.188·28-s − 0.588·29-s + 0.516·30-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: Trivial
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 \( 1 - 2.82T + 3T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 - 3.17T + 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 - 5.17T + 79T^{2} \)
83 \( 1 - 2.34T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 0.828T + 97T^{2} \)
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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

−7.69290706061046991419729170522, −7.12410196158094950650297801529, −6.41715937623756269132467792109, −5.27176126435502999331733538728, −4.33114404947085658146874566075, −3.78682559478378396781756919967, −2.82989658276470805211087713559, −2.25843032292322031240915289699, −1.50404889698204852610377958237, 0, 1.50404889698204852610377958237, 2.25843032292322031240915289699, 2.82989658276470805211087713559, 3.78682559478378396781756919967, 4.33114404947085658146874566075, 5.27176126435502999331733538728, 6.41715937623756269132467792109, 7.12410196158094950650297801529, 7.69290706061046991419729170522

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