Properties

Label 2-8470-1.1-c1-0-129
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 + 3.24·3-s + 4-s − 5-s + 3.24·6-s − 7-s + 8-s + 7.51·9-s − 10-s + 3.24·12-s + 2.14·13-s − 14-s − 3.24·15-s + 16-s + 1.43·17-s + 7.51·18-s − 0.633·19-s − 20-s − 3.24·21-s − 5.61·23-s + 3.24·24-s + 25-s + 2.14·26-s + 14.6·27-s − 28-s − 1.19·29-s − 3.24·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.87·3-s + 0.5·4-s − 0.447·5-s + 1.32·6-s − 0.377·7-s + 0.353·8-s + 2.50·9-s − 0.316·10-s + 0.936·12-s + 0.594·13-s − 0.267·14-s − 0.837·15-s + 0.250·16-s + 0.349·17-s + 1.77·18-s − 0.145·19-s − 0.223·20-s − 0.707·21-s − 1.17·23-s + 0.662·24-s + 0.200·25-s + 0.420·26-s + 2.82·27-s − 0.188·28-s − 0.222·29-s − 0.592·30-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

\(L(1)\) \(\approx\) \(6.750192342\)
\(L(\frac12)\) \(\approx\) \(6.750192342\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.24T + 3T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + 0.633T + 19T^{2} \)
23 \( 1 + 5.61T + 23T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 - 6.95T + 31T^{2} \)
37 \( 1 - 4.80T + 37T^{2} \)
41 \( 1 + 8.93T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 - 8.29T + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 + 1.19T + 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 - 6.43T + 73T^{2} \)
79 \( 1 + 7.69T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 2.96T + 97T^{2} \)
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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

−7.73031826076976371811516951027, −7.33370349803931097474681230259, −6.48172558893643160548671484372, −5.77947592255442018270122925932, −4.59322499709738341222269943237, −4.02628357656133709036580418006, −3.53469327639876763298889895799, −2.77887181567991881106630115922, −2.17385134441837913622240390129, −1.10290541643892074970516549485, 1.10290541643892074970516549485, 2.17385134441837913622240390129, 2.77887181567991881106630115922, 3.53469327639876763298889895799, 4.02628357656133709036580418006, 4.59322499709738341222269943237, 5.77947592255442018270122925932, 6.48172558893643160548671484372, 7.33370349803931097474681230259, 7.73031826076976371811516951027

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