Properties

Label 2-8470-1.1-c1-0-9
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 + 0.618·3-s + 4-s − 5-s − 0.618·6-s + 7-s − 8-s − 2.61·9-s + 10-s + 0.618·12-s − 5.23·13-s − 14-s − 0.618·15-s + 16-s − 1.61·17-s + 2.61·18-s − 0.381·19-s − 20-s + 0.618·21-s − 8.47·23-s − 0.618·24-s + 25-s + 5.23·26-s − 3.47·27-s + 28-s + 4·29-s + 0.618·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.356·3-s + 0.5·4-s − 0.447·5-s − 0.252·6-s + 0.377·7-s − 0.353·8-s − 0.872·9-s + 0.316·10-s + 0.178·12-s − 1.45·13-s − 0.267·14-s − 0.159·15-s + 0.250·16-s − 0.392·17-s + 0.617·18-s − 0.0876·19-s − 0.223·20-s + 0.134·21-s − 1.76·23-s − 0.126·24-s + 0.200·25-s + 1.02·26-s − 0.668·27-s + 0.188·28-s + 0.742·29-s + 0.112·30-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

\(L(1)\) \(\approx\) \(0.7065494553\)
\(L(\frac12)\) \(\approx\) \(0.7065494553\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618T + 3T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 + 0.381T + 19T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 4.47T + 31T^{2} \)
37 \( 1 - 9.70T + 37T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + 6.76T + 47T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 + 3.61T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 - 4.32T + 73T^{2} \)
79 \( 1 - 1.70T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 9.85T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−7.86313493333606840239431088033, −7.47048303237400998249831321071, −6.45905101675920342266162101569, −5.90185869858026475314808896363, −4.93017822102945869922969412424, −4.28481647097184296036985897275, −3.27500799836459067882814022239, −2.50446520073044682072894822147, −1.87309826651203929108742451742, −0.42389219292443958376726630680, 0.42389219292443958376726630680, 1.87309826651203929108742451742, 2.50446520073044682072894822147, 3.27500799836459067882814022239, 4.28481647097184296036985897275, 4.93017822102945869922969412424, 5.90185869858026475314808896363, 6.45905101675920342266162101569, 7.47048303237400998249831321071, 7.86313493333606840239431088033

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