Properties

Label 2-8470-1.1-c1-0-69
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 + 0.381·3-s + 4-s − 5-s + 0.381·6-s + 7-s + 8-s − 2.85·9-s − 10-s + 0.381·12-s + 2·13-s + 14-s − 0.381·15-s + 16-s + 0.618·17-s − 2.85·18-s + 0.145·19-s − 20-s + 0.381·21-s + 6·23-s + 0.381·24-s + 25-s + 2·26-s − 2.23·27-s + 28-s − 1.23·29-s − 0.381·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.220·3-s + 0.5·4-s − 0.447·5-s + 0.155·6-s + 0.377·7-s + 0.353·8-s − 0.951·9-s − 0.316·10-s + 0.110·12-s + 0.554·13-s + 0.267·14-s − 0.0986·15-s + 0.250·16-s + 0.149·17-s − 0.672·18-s + 0.0334·19-s − 0.223·20-s + 0.0833·21-s + 1.25·23-s + 0.0779·24-s + 0.200·25-s + 0.392·26-s − 0.430·27-s + 0.188·28-s − 0.229·29-s − 0.0697·30-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: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.332926674\)
\(L(\frac12)\) \(\approx\) \(3.332926674\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.381T + 3T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 0.618T + 17T^{2} \)
19 \( 1 - 0.145T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 1.23T + 29T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 - 5.32T + 41T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 + 3.23T + 53T^{2} \)
59 \( 1 + 5.38T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 - 3.70T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 - 0.472T + 79T^{2} \)
83 \( 1 + 5.32T + 83T^{2} \)
89 \( 1 - 1.90T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.74375263481553139393716318327, −7.12516277381347191447011872038, −6.23974685029966967146232500651, −5.73201778475460812888489047479, −4.89580090978530951344422081756, −4.33915818886218504143875291355, −3.34961031769658516483121276259, −2.95760442444676287034331369049, −1.93047527555627234158601924663, −0.796403716217462582161886943804, 0.796403716217462582161886943804, 1.93047527555627234158601924663, 2.95760442444676287034331369049, 3.34961031769658516483121276259, 4.33915818886218504143875291355, 4.89580090978530951344422081756, 5.73201778475460812888489047479, 6.23974685029966967146232500651, 7.12516277381347191447011872038, 7.74375263481553139393716318327

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