Properties

Label 2-8470-1.1-c1-0-141
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 − 0.418·3-s + 4-s + 5-s + 0.418·6-s − 7-s − 8-s − 2.82·9-s − 10-s − 0.418·12-s + 5.30·13-s + 14-s − 0.418·15-s + 16-s − 1.47·17-s + 2.82·18-s − 0.952·19-s + 20-s + 0.418·21-s + 2.37·23-s + 0.418·24-s + 25-s − 5.30·26-s + 2.43·27-s − 28-s − 5.97·29-s + 0.418·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.241·3-s + 0.5·4-s + 0.447·5-s + 0.171·6-s − 0.377·7-s − 0.353·8-s − 0.941·9-s − 0.316·10-s − 0.120·12-s + 1.47·13-s + 0.267·14-s − 0.108·15-s + 0.250·16-s − 0.358·17-s + 0.665·18-s − 0.218·19-s + 0.223·20-s + 0.0914·21-s + 0.495·23-s + 0.0855·24-s + 0.200·25-s − 1.03·26-s + 0.469·27-s − 0.188·28-s − 1.11·29-s + 0.0764·30-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 \( 1 + 0.418T + 3T^{2} \)
13 \( 1 - 5.30T + 13T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 + 0.952T + 19T^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 + 5.97T + 29T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
37 \( 1 - 0.0224T + 37T^{2} \)
41 \( 1 - 1.53T + 41T^{2} \)
43 \( 1 + 3.83T + 43T^{2} \)
47 \( 1 + 1.50T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 - 0.768T + 59T^{2} \)
61 \( 1 + 8.06T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 7.28T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 6.13T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−7.53206709471357863644512527863, −6.58315514364810615425262181109, −6.20905748092533099970048839580, −5.62994009779115965708311334764, −4.76076565541227259024772181535, −3.63926556877605486613028881474, −3.02697828856266508585647460515, −2.07357966365610978109884131407, −1.14365062572257572899502140903, 0, 1.14365062572257572899502140903, 2.07357966365610978109884131407, 3.02697828856266508585647460515, 3.63926556877605486613028881474, 4.76076565541227259024772181535, 5.62994009779115965708311334764, 6.20905748092533099970048839580, 6.58315514364810615425262181109, 7.53206709471357863644512527863

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