Properties

Label 2-8470-1.1-c1-0-184
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 + 2·3-s + 4-s + 5-s − 2·6-s − 7-s − 8-s + 9-s − 10-s + 2·12-s − 6.74·13-s + 14-s + 2·15-s + 16-s + 6.74·17-s − 18-s + 6.74·19-s + 20-s − 2·21-s − 6.74·23-s − 2·24-s + 25-s + 6.74·26-s − 4·27-s − 28-s − 8.74·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.577·12-s − 1.87·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s + 1.63·17-s − 0.235·18-s + 1.54·19-s + 0.223·20-s − 0.436·21-s − 1.40·23-s − 0.408·24-s + 0.200·25-s + 1.32·26-s − 0.769·27-s − 0.188·28-s − 1.62·29-s − 0.365·30-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 \( 1 - 2T + 3T^{2} \)
13 \( 1 + 6.74T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 - 0.744T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 1.25T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
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

−7.59578488362350570607633471147, −7.24764137113673460524025099695, −6.00385949210381561133396615794, −5.58154166565594003828796728812, −4.60319927087410754035859961774, −3.43848688048706259786020070583, −2.98818382239517239835043381242, −2.24862362999248038185364216190, −1.41407636466437536620068272721, 0, 1.41407636466437536620068272721, 2.24862362999248038185364216190, 2.98818382239517239835043381242, 3.43848688048706259786020070583, 4.60319927087410754035859961774, 5.58154166565594003828796728812, 6.00385949210381561133396615794, 7.24764137113673460524025099695, 7.59578488362350570607633471147

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