Properties

Label 2-8470-1.1-c1-0-20
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 − 0.618·3-s + 4-s − 5-s + 0.618·6-s + 7-s − 8-s − 2.61·9-s + 10-s − 0.618·12-s − 0.763·13-s − 14-s + 0.618·15-s + 16-s + 0.381·17-s + 2.61·18-s + 1.14·19-s − 20-s − 0.618·21-s + 8.47·23-s + 0.618·24-s + 25-s + 0.763·26-s + 3.47·27-s + 28-s − 4.47·29-s − 0.618·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.356·3-s + 0.5·4-s − 0.447·5-s + 0.252·6-s + 0.377·7-s − 0.353·8-s − 0.872·9-s + 0.316·10-s − 0.178·12-s − 0.211·13-s − 0.267·14-s + 0.159·15-s + 0.250·16-s + 0.0926·17-s + 0.617·18-s + 0.262·19-s − 0.223·20-s − 0.134·21-s + 1.76·23-s + 0.126·24-s + 0.200·25-s + 0.149·26-s + 0.668·27-s + 0.188·28-s − 0.830·29-s − 0.112·30-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

\(L(1)\) \(\approx\) \(0.8253465269\)
\(L(\frac12)\) \(\approx\) \(0.8253465269\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 3T^{2} \)
13 \( 1 + 0.763T + 13T^{2} \)
17 \( 1 - 0.381T + 17T^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 + 6.76T + 37T^{2} \)
41 \( 1 + 3.85T + 41T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 - 9.23T + 47T^{2} \)
53 \( 1 + 7.23T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 0.291T + 61T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + 6.94T + 71T^{2} \)
73 \( 1 + 3.85T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 2.14T + 83T^{2} \)
89 \( 1 + 8.14T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−7.69873217908466397607728991657, −7.25553467154901869464876109397, −6.61277221642534947516244448214, −5.52951114911103229183786782461, −5.33414615019700093194926103879, −4.25350862504594967747850334154, −3.32254418999150443307619818676, −2.62881031823272312295719132076, −1.55111260496128532395781375442, −0.51442012573373620441364043649, 0.51442012573373620441364043649, 1.55111260496128532395781375442, 2.62881031823272312295719132076, 3.32254418999150443307619818676, 4.25350862504594967747850334154, 5.33414615019700093194926103879, 5.52951114911103229183786782461, 6.61277221642534947516244448214, 7.25553467154901869464876109397, 7.69873217908466397607728991657

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