Properties

Label 2-8470-1.1-c1-0-166
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 + 1.91·3-s + 4-s − 5-s − 1.91·6-s − 7-s − 8-s + 0.683·9-s + 10-s + 1.91·12-s − 0.869·13-s + 14-s − 1.91·15-s + 16-s − 4.15·17-s − 0.683·18-s + 7.56·19-s − 20-s − 1.91·21-s − 2·23-s − 1.91·24-s + 25-s + 0.869·26-s − 4.44·27-s − 28-s − 1.19·29-s + 1.91·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.10·3-s + 0.5·4-s − 0.447·5-s − 0.783·6-s − 0.377·7-s − 0.353·8-s + 0.227·9-s + 0.316·10-s + 0.554·12-s − 0.241·13-s + 0.267·14-s − 0.495·15-s + 0.250·16-s − 1.00·17-s − 0.160·18-s + 1.73·19-s − 0.223·20-s − 0.418·21-s − 0.417·23-s − 0.391·24-s + 0.200·25-s + 0.170·26-s − 0.855·27-s − 0.188·28-s − 0.222·29-s + 0.350·30-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 \( 1 - 1.91T + 3T^{2} \)
13 \( 1 + 0.869T + 13T^{2} \)
17 \( 1 + 4.15T + 17T^{2} \)
19 \( 1 - 7.56T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 + 9.86T + 41T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.57T + 53T^{2} \)
59 \( 1 + 7.43T + 59T^{2} \)
61 \( 1 + 7.73T + 61T^{2} \)
67 \( 1 - 3.56T + 67T^{2} \)
71 \( 1 + 2.23T + 71T^{2} \)
73 \( 1 - 17.0T + 73T^{2} \)
79 \( 1 + 3.13T + 79T^{2} \)
83 \( 1 + 8.22T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−7.64827139422441836107861321580, −7.01332139697570283957541823408, −6.31467788723494188413836237421, −5.40874795183790324590013976173, −4.47141800036798768317385385863, −3.59686935022065176530043590751, −2.94977437651244492772893670245, −2.35008290452733810692599701387, −1.25639945374334753573577170555, 0, 1.25639945374334753573577170555, 2.35008290452733810692599701387, 2.94977437651244492772893670245, 3.59686935022065176530043590751, 4.47141800036798768317385385863, 5.40874795183790324590013976173, 6.31467788723494188413836237421, 7.01332139697570283957541823408, 7.64827139422441836107861321580

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