Properties

Label 2-8470-1.1-c1-0-18
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 − 2.24·3-s + 4-s + 5-s + 2.24·6-s + 7-s − 8-s + 2.05·9-s − 10-s − 2.24·12-s − 0.941·13-s − 14-s − 2.24·15-s + 16-s − 6.49·17-s − 2.05·18-s + 4.36·19-s + 20-s − 2.24·21-s + 6.24·23-s + 2.24·24-s + 25-s + 0.941·26-s + 2.11·27-s + 28-s − 8.74·29-s + 2.24·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s + 0.918·6-s + 0.377·7-s − 0.353·8-s + 0.686·9-s − 0.316·10-s − 0.649·12-s − 0.261·13-s − 0.267·14-s − 0.580·15-s + 0.250·16-s − 1.57·17-s − 0.485·18-s + 1.00·19-s + 0.223·20-s − 0.490·21-s + 1.30·23-s + 0.459·24-s + 0.200·25-s + 0.184·26-s + 0.407·27-s + 0.188·28-s − 1.62·29-s + 0.410·30-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

\(L(1)\) \(\approx\) \(0.6923054243\)
\(L(\frac12)\) \(\approx\) \(0.6923054243\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.24T + 3T^{2} \)
13 \( 1 + 0.941T + 13T^{2} \)
17 \( 1 + 6.49T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 9.55T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 + 7.67T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 4.74T + 53T^{2} \)
59 \( 1 + 1.88T + 59T^{2} \)
61 \( 1 + 9.11T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.48695660552970232118041461455, −7.19558674327580512944838775717, −6.40966989065172094313639560424, −5.74906769180087884411473950460, −5.19462346015292754778413417257, −4.55204984954478532663162877535, −3.42654582609784171412374442967, −2.34779885784852404045892139007, −1.53216886223004944863851438552, −0.49752383748670777740194383243, 0.49752383748670777740194383243, 1.53216886223004944863851438552, 2.34779885784852404045892139007, 3.42654582609784171412374442967, 4.55204984954478532663162877535, 5.19462346015292754778413417257, 5.74906769180087884411473950460, 6.40966989065172094313639560424, 7.19558674327580512944838775717, 7.48695660552970232118041461455

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