Properties

Label 2-8470-1.1-c1-0-214
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3.25·3-s + 4-s + 5-s − 3.25·6-s − 7-s − 8-s + 7.59·9-s − 10-s + 3.25·12-s − 3.57·13-s + 14-s + 3.25·15-s + 16-s − 3.01·17-s − 7.59·18-s − 2.86·19-s + 20-s − 3.25·21-s − 9.00·23-s − 3.25·24-s + 25-s + 3.57·26-s + 14.9·27-s − 28-s − 1.56·29-s − 3.25·30-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 + 2.53·9-s − 0.316·10-s + 0.939·12-s − 0.992·13-s + 0.267·14-s + 0.840·15-s + 0.250·16-s − 0.731·17-s − 1.79·18-s − 0.657·19-s + 0.223·20-s − 0.710·21-s − 1.87·23-s − 0.664·24-s + 0.200·25-s + 0.701·26-s + 2.87·27-s − 0.188·28-s − 0.290·29-s − 0.594·30-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: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.25T + 3T^{2} \)
13 \( 1 + 3.57T + 13T^{2} \)
17 \( 1 + 3.01T + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
23 \( 1 + 9.00T + 23T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 + 0.473T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 3.84T + 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + 9.61T + 47T^{2} \)
53 \( 1 + 5.62T + 53T^{2} \)
59 \( 1 + 2.73T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 0.752T + 67T^{2} \)
71 \( 1 + 1.90T + 71T^{2} \)
73 \( 1 + 1.66T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + 1.05T + 97T^{2} \)
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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

−7.73038433585539069375883949262, −6.83714747147031479765526980404, −6.57332677314229326588098468494, −5.33807304746527400040063802625, −4.36872224711301121131091465194, −3.63363041504930315044552060629, −2.88299472200833626955013392851, −2.05071609056907941882022980827, −1.78245011799957503055671476228, 0, 1.78245011799957503055671476228, 2.05071609056907941882022980827, 2.88299472200833626955013392851, 3.63363041504930315044552060629, 4.36872224711301121131091465194, 5.33807304746527400040063802625, 6.57332677314229326588098468494, 6.83714747147031479765526980404, 7.73038433585539069375883949262

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