Properties

Label 2-8470-1.1-c1-0-105
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 + 1.14·3-s + 4-s + 5-s + 1.14·6-s + 7-s + 8-s − 1.68·9-s + 10-s + 1.14·12-s + 14-s + 1.14·15-s + 16-s − 2.68·17-s − 1.68·18-s + 5.53·19-s + 20-s + 1.14·21-s − 1.83·23-s + 1.14·24-s + 25-s − 5.37·27-s + 28-s + 3.83·29-s + 1.14·30-s + 9.66·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.661·3-s + 0.5·4-s + 0.447·5-s + 0.468·6-s + 0.377·7-s + 0.353·8-s − 0.561·9-s + 0.316·10-s + 0.330·12-s + 0.267·14-s + 0.295·15-s + 0.250·16-s − 0.651·17-s − 0.397·18-s + 1.27·19-s + 0.223·20-s + 0.250·21-s − 0.382·23-s + 0.234·24-s + 0.200·25-s − 1.03·27-s + 0.188·28-s + 0.711·29-s + 0.209·30-s + 1.73·31-s + 0.176·32-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

\(L(1)\) \(\approx\) \(4.896851939\)
\(L(\frac12)\) \(\approx\) \(4.896851939\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.14T + 3T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 - 3.83T + 29T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 + 1.83T + 37T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 - 4.97T + 43T^{2} \)
47 \( 1 + 0.393T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 0.685T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 - 3.37T + 67T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + 8.97T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 9.37T + 83T^{2} \)
89 \( 1 - 6.58T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.74158337071030412704895476804, −7.10370111792033627629832926597, −6.27435434691687912970198636475, −5.68757453185233107641659273540, −4.97655566258029294224382771120, −4.27832068138821671020105535832, −3.39147442546807728218434604590, −2.69746653862613125457301363187, −2.10200146146536315585326498236, −0.959195220023369098798078773193, 0.959195220023369098798078773193, 2.10200146146536315585326498236, 2.69746653862613125457301363187, 3.39147442546807728218434604590, 4.27832068138821671020105535832, 4.97655566258029294224382771120, 5.68757453185233107641659273540, 6.27435434691687912970198636475, 7.10370111792033627629832926597, 7.74158337071030412704895476804

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