Properties

Label 2-8470-1.1-c1-0-6
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 − 3.40·3-s + 4-s − 5-s − 3.40·6-s − 7-s + 8-s + 8.61·9-s − 10-s − 3.40·12-s − 3.69·13-s − 14-s + 3.40·15-s + 16-s − 3.63·17-s + 8.61·18-s + 2.01·19-s − 20-s + 3.40·21-s − 5.90·23-s − 3.40·24-s + 25-s − 3.69·26-s − 19.1·27-s − 28-s − 2.77·29-s + 3.40·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.96·3-s + 0.5·4-s − 0.447·5-s − 1.39·6-s − 0.377·7-s + 0.353·8-s + 2.87·9-s − 0.316·10-s − 0.984·12-s − 1.02·13-s − 0.267·14-s + 0.880·15-s + 0.250·16-s − 0.880·17-s + 2.03·18-s + 0.462·19-s − 0.223·20-s + 0.743·21-s − 1.23·23-s − 0.695·24-s + 0.200·25-s − 0.724·26-s − 3.68·27-s − 0.188·28-s − 0.515·29-s + 0.622·30-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

\(L(1)\) \(\approx\) \(0.7154320367\)
\(L(\frac12)\) \(\approx\) \(0.7154320367\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.40T + 3T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 + 3.63T + 17T^{2} \)
19 \( 1 - 2.01T + 19T^{2} \)
23 \( 1 + 5.90T + 23T^{2} \)
29 \( 1 + 2.77T + 29T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 - 3.22T + 37T^{2} \)
41 \( 1 + 8.13T + 41T^{2} \)
43 \( 1 + 3.87T + 43T^{2} \)
47 \( 1 - 9.67T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 + 1.83T + 73T^{2} \)
79 \( 1 - 2.95T + 79T^{2} \)
83 \( 1 + 8.37T + 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−7.33495417217510428354838460041, −6.90006444619394038944575127124, −6.35888391175809472211848668960, −5.51595124610418683896476286475, −5.23351995615013903815024332888, −4.18235110208384910377742034114, −4.09835082402861837956307381079, −2.68237276578320053557581459210, −1.63083676721619656295848122034, −0.40958876409728795275964751965, 0.40958876409728795275964751965, 1.63083676721619656295848122034, 2.68237276578320053557581459210, 4.09835082402861837956307381079, 4.18235110208384910377742034114, 5.23351995615013903815024332888, 5.51595124610418683896476286475, 6.35888391175809472211848668960, 6.90006444619394038944575127124, 7.33495417217510428354838460041

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