Properties

Label 2-8470-1.1-c1-0-58
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 − 1.12·3-s + 4-s − 5-s − 1.12·6-s − 7-s + 8-s − 1.73·9-s − 10-s − 1.12·12-s + 7.06·13-s − 14-s + 1.12·15-s + 16-s + 3.44·17-s − 1.73·18-s + 3.43·19-s − 20-s + 1.12·21-s + 3.63·23-s − 1.12·24-s + 25-s + 7.06·26-s + 5.32·27-s − 28-s − 8.69·29-s + 1.12·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.649·3-s + 0.5·4-s − 0.447·5-s − 0.459·6-s − 0.377·7-s + 0.353·8-s − 0.578·9-s − 0.316·10-s − 0.324·12-s + 1.96·13-s − 0.267·14-s + 0.290·15-s + 0.250·16-s + 0.835·17-s − 0.409·18-s + 0.788·19-s − 0.223·20-s + 0.245·21-s + 0.758·23-s − 0.229·24-s + 0.200·25-s + 1.38·26-s + 1.02·27-s − 0.188·28-s − 1.61·29-s + 0.205·30-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

\(L(1)\) \(\approx\) \(2.311653251\)
\(L(\frac12)\) \(\approx\) \(2.311653251\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.12T + 3T^{2} \)
13 \( 1 - 7.06T + 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 + 8.69T + 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + 0.780T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 + 5.39T + 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 0.456T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 6.81T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 - 7.40T + 83T^{2} \)
89 \( 1 + 4.97T + 89T^{2} \)
97 \( 1 - 2.88T + 97T^{2} \)
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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

−7.49576665983236155610311614894, −7.07460504505462032723544427392, −6.10388443025480752005739752082, −5.73538905213925921726061938696, −5.22076550721770691285811924464, −4.18426113991224509281492031676, −3.43835220155321323055394260576, −3.10083032428550460206026454233, −1.68116901702903111545204043288, −0.71248825117380874315896609022, 0.71248825117380874315896609022, 1.68116901702903111545204043288, 3.10083032428550460206026454233, 3.43835220155321323055394260576, 4.18426113991224509281492031676, 5.22076550721770691285811924464, 5.73538905213925921726061938696, 6.10388443025480752005739752082, 7.07460504505462032723544427392, 7.49576665983236155610311614894

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