Properties

Label 2-8470-1.1-c1-0-194
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 + 2.44·3-s + 4-s + 5-s − 2.44·6-s − 7-s − 8-s + 2.97·9-s − 10-s + 2.44·12-s − 1.00·13-s + 14-s + 2.44·15-s + 16-s − 0.974·17-s − 2.97·18-s − 5.13·19-s + 20-s − 2.44·21-s − 4.97·23-s − 2.44·24-s + 25-s + 1.00·26-s − 0.0527·27-s − 28-s − 2.89·29-s − 2.44·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.447·5-s − 0.998·6-s − 0.377·7-s − 0.353·8-s + 0.992·9-s − 0.316·10-s + 0.705·12-s − 0.278·13-s + 0.267·14-s + 0.631·15-s + 0.250·16-s − 0.236·17-s − 0.702·18-s − 1.17·19-s + 0.223·20-s − 0.533·21-s − 1.03·23-s − 0.499·24-s + 0.200·25-s + 0.196·26-s − 0.0101·27-s − 0.188·28-s − 0.537·29-s − 0.446·30-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 \( 1 - 2.44T + 3T^{2} \)
13 \( 1 + 1.00T + 13T^{2} \)
17 \( 1 + 0.974T + 17T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + 2.89T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 - 9.79T + 37T^{2} \)
41 \( 1 + 3.10T + 41T^{2} \)
43 \( 1 - 9.46T + 43T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 - 3.98T + 53T^{2} \)
59 \( 1 + 0.902T + 59T^{2} \)
61 \( 1 + 7.11T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 - 2.21T + 71T^{2} \)
73 \( 1 - 4.41T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 8.95T + 83T^{2} \)
89 \( 1 + 5.28T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−7.60057488702586050833333495542, −7.04932183498175893834361219165, −6.16081450587214588068526047685, −5.61522079797374862710282012092, −4.28452637516382305211332987534, −3.79598810142175860538031767559, −2.68765335910088442537706570873, −2.35575016827317682901191023821, −1.49261623687931561200443730167, 0, 1.49261623687931561200443730167, 2.35575016827317682901191023821, 2.68765335910088442537706570873, 3.79598810142175860538031767559, 4.28452637516382305211332987534, 5.61522079797374862710282012092, 6.16081450587214588068526047685, 7.04932183498175893834361219165, 7.60057488702586050833333495542

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