Properties

Label 2-8470-1.1-c1-0-53
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 − 1.68·3-s + 4-s + 5-s − 1.68·6-s − 7-s + 8-s − 0.154·9-s + 10-s − 1.68·12-s − 0.0425·13-s − 14-s − 1.68·15-s + 16-s + 4.72·17-s − 0.154·18-s + 1.71·19-s + 20-s + 1.68·21-s − 6.95·23-s − 1.68·24-s + 25-s − 0.0425·26-s + 5.32·27-s − 28-s + 1.34·29-s − 1.68·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.973·3-s + 0.5·4-s + 0.447·5-s − 0.688·6-s − 0.377·7-s + 0.353·8-s − 0.0514·9-s + 0.316·10-s − 0.486·12-s − 0.0118·13-s − 0.267·14-s − 0.435·15-s + 0.250·16-s + 1.14·17-s − 0.0363·18-s + 0.393·19-s + 0.223·20-s + 0.368·21-s − 1.45·23-s − 0.344·24-s + 0.200·25-s − 0.00834·26-s + 1.02·27-s − 0.188·28-s + 0.249·29-s − 0.307·30-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

\(L(1)\) \(\approx\) \(2.210406998\)
\(L(\frac12)\) \(\approx\) \(2.210406998\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.68T + 3T^{2} \)
13 \( 1 + 0.0425T + 13T^{2} \)
17 \( 1 - 4.72T + 17T^{2} \)
19 \( 1 - 1.71T + 19T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 - 1.34T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 3.91T + 41T^{2} \)
43 \( 1 + 6.42T + 43T^{2} \)
47 \( 1 - 4.59T + 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 1.98T + 73T^{2} \)
79 \( 1 - 8.87T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 2.93T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−7.78471531827781674216789124522, −6.70602702613022865935704283486, −6.19014255594198576311851881999, −5.83575129502058173080393898736, −5.05346253870865437533479274952, −4.50976457066452219525786613816, −3.45217129864692571215120697000, −2.84553053769287833215163476179, −1.76474703321371671886084958456, −0.68861058607790811979049841117, 0.68861058607790811979049841117, 1.76474703321371671886084958456, 2.84553053769287833215163476179, 3.45217129864692571215120697000, 4.50976457066452219525786613816, 5.05346253870865437533479274952, 5.83575129502058173080393898736, 6.19014255594198576311851881999, 6.70602702613022865935704283486, 7.78471531827781674216789124522

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