Properties

Label 2-8470-1.1-c1-0-215
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 + 2.16·3-s + 4-s − 5-s + 2.16·6-s + 7-s + 8-s + 1.67·9-s − 10-s + 2.16·12-s − 3.92·13-s + 14-s − 2.16·15-s + 16-s − 6.06·17-s + 1.67·18-s − 5.77·19-s − 20-s + 2.16·21-s + 1.51·23-s + 2.16·24-s + 25-s − 3.92·26-s − 2.85·27-s + 28-s − 4.12·29-s − 2.16·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.24·3-s + 0.5·4-s − 0.447·5-s + 0.883·6-s + 0.377·7-s + 0.353·8-s + 0.559·9-s − 0.316·10-s + 0.624·12-s − 1.08·13-s + 0.267·14-s − 0.558·15-s + 0.250·16-s − 1.46·17-s + 0.395·18-s − 1.32·19-s − 0.223·20-s + 0.472·21-s + 0.315·23-s + 0.441·24-s + 0.200·25-s − 0.768·26-s − 0.549·27-s + 0.188·28-s − 0.765·29-s − 0.394·30-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 \( 1 - 2.16T + 3T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
17 \( 1 + 6.06T + 17T^{2} \)
19 \( 1 + 5.77T + 19T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 + 4.12T + 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + 5.54T + 41T^{2} \)
43 \( 1 - 6.51T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 + 3.77T + 59T^{2} \)
61 \( 1 + 0.532T + 61T^{2} \)
67 \( 1 + 1.15T + 67T^{2} \)
71 \( 1 - 0.623T + 71T^{2} \)
73 \( 1 - 3.10T + 73T^{2} \)
79 \( 1 - 3.76T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−7.37535668303322064249956052852, −6.91548140592868209472038671668, −6.13147738618648311913108579757, −5.07508238351369037109191873153, −4.48707477359908688435348497577, −3.92486906127401289644781623735, −3.03935349662536666867092087927, −2.37934078168086316022645722123, −1.78987209666502174251391932816, 0, 1.78987209666502174251391932816, 2.37934078168086316022645722123, 3.03935349662536666867092087927, 3.92486906127401289644781623735, 4.48707477359908688435348497577, 5.07508238351369037109191873153, 6.13147738618648311913108579757, 6.91548140592868209472038671668, 7.37535668303322064249956052852

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