Properties

Label 2-8470-1.1-c1-0-84
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 − 2.54·3-s + 4-s + 5-s + 2.54·6-s − 7-s − 8-s + 3.46·9-s − 10-s − 2.54·12-s + 6.42·13-s + 14-s − 2.54·15-s + 16-s + 3.95·17-s − 3.46·18-s − 2.46·19-s + 20-s + 2.54·21-s + 4.99·23-s + 2.54·24-s + 25-s − 6.42·26-s − 1.19·27-s − 28-s + 9.04·29-s + 2.54·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.46·3-s + 0.5·4-s + 0.447·5-s + 1.03·6-s − 0.377·7-s − 0.353·8-s + 1.15·9-s − 0.316·10-s − 0.734·12-s + 1.78·13-s + 0.267·14-s − 0.656·15-s + 0.250·16-s + 0.958·17-s − 0.817·18-s − 0.566·19-s + 0.223·20-s + 0.555·21-s + 1.04·23-s + 0.519·24-s + 0.200·25-s − 1.25·26-s − 0.229·27-s − 0.188·28-s + 1.67·29-s + 0.464·30-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

\(L(1)\) \(\approx\) \(1.243599014\)
\(L(\frac12)\) \(\approx\) \(1.243599014\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.54T + 3T^{2} \)
13 \( 1 - 6.42T + 13T^{2} \)
17 \( 1 - 3.95T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 - 9.04T + 29T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 6.95T + 41T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + 8.37T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 - 3.59T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 5.90T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 - 6.21T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.85335786996239964040218229616, −6.80488891643553997769595981638, −6.22209931779155718503777502254, −6.13775560637254531366145689659, −5.18853095554908693437829449492, −4.45977790513896467129545797671, −3.41564152627358228739264780078, −2.53482307271622216214480836501, −1.08732303067700030086539952349, −0.878978692381912878813734382724, 0.878978692381912878813734382724, 1.08732303067700030086539952349, 2.53482307271622216214480836501, 3.41564152627358228739264780078, 4.45977790513896467129545797671, 5.18853095554908693437829449492, 6.13775560637254531366145689659, 6.22209931779155718503777502254, 6.80488891643553997769595981638, 7.85335786996239964040218229616

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