Properties

Label 2-8470-1.1-c1-0-24
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 − 0.803·3-s + 4-s + 5-s + 0.803·6-s + 7-s − 8-s − 2.35·9-s − 10-s − 0.803·12-s + 2.64·13-s − 14-s − 0.803·15-s + 16-s − 1.60·17-s + 2.35·18-s − 1.90·19-s + 20-s − 0.803·21-s − 4.22·23-s + 0.803·24-s + 25-s − 2.64·26-s + 4.30·27-s + 28-s − 8.97·29-s + 0.803·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.463·3-s + 0.5·4-s + 0.447·5-s + 0.327·6-s + 0.377·7-s − 0.353·8-s − 0.784·9-s − 0.316·10-s − 0.231·12-s + 0.733·13-s − 0.267·14-s − 0.207·15-s + 0.250·16-s − 0.389·17-s + 0.554·18-s − 0.438·19-s + 0.223·20-s − 0.175·21-s − 0.881·23-s + 0.163·24-s + 0.200·25-s − 0.518·26-s + 0.827·27-s + 0.188·28-s − 1.66·29-s + 0.146·30-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

\(L(1)\) \(\approx\) \(1.007297333\)
\(L(\frac12)\) \(\approx\) \(1.007297333\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.803T + 3T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + 1.60T + 17T^{2} \)
19 \( 1 + 1.90T + 19T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 5.60T + 37T^{2} \)
41 \( 1 - 9.69T + 41T^{2} \)
43 \( 1 - 4.00T + 43T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 4.16T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 4.58T + 71T^{2} \)
73 \( 1 + 3.69T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 3.41T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−7.85436319763499270042186323355, −7.17890499014761046249885097539, −6.22007155745277315136158798337, −5.96700159643613654514597828498, −5.22277828190978638381362459153, −4.27955351905739271939631447235, −3.39496131615124454827552461909, −2.38188295533834072831087588333, −1.70084507219938601940677700067, −0.55481135790589426225752656456, 0.55481135790589426225752656456, 1.70084507219938601940677700067, 2.38188295533834072831087588333, 3.39496131615124454827552461909, 4.27955351905739271939631447235, 5.22277828190978638381362459153, 5.96700159643613654514597828498, 6.22007155745277315136158798337, 7.17890499014761046249885097539, 7.85436319763499270042186323355

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