Properties

Label 2-8470-1.1-c1-0-40
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 − 4.68·13-s − 14-s + 1.14·15-s + 16-s + 0.292·17-s + 1.68·18-s − 6.51·19-s + 20-s + 1.14·21-s + 2.85·23-s − 1.14·24-s + 25-s + 4.68·26-s − 5.37·27-s + 28-s + 1.43·29-s − 1.14·30-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 − 1.29·13-s − 0.267·14-s + 0.295·15-s + 0.250·16-s + 0.0709·17-s + 0.397·18-s − 1.49·19-s + 0.223·20-s + 0.250·21-s + 0.595·23-s − 0.234·24-s + 0.200·25-s + 0.918·26-s − 1.03·27-s + 0.188·28-s + 0.267·29-s − 0.209·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.596359104\)
\(L(\frac12)\) \(\approx\) \(1.596359104\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.68T + 13T^{2} \)
17 \( 1 - 0.292T + 17T^{2} \)
19 \( 1 + 6.51T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 - 1.43T + 29T^{2} \)
31 \( 1 - 0.978T + 31T^{2} \)
37 \( 1 - 0.853T + 37T^{2} \)
41 \( 1 - 6.22T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 9.95T + 47T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 0.585T + 67T^{2} \)
71 \( 1 + 0.335T + 71T^{2} \)
73 \( 1 - 3.70T + 73T^{2} \)
79 \( 1 - 2.51T + 79T^{2} \)
83 \( 1 - 1.70T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−7.903712515451102369404979111736, −7.31419616245497645615733726648, −6.52140951385644753733343816993, −5.84907127756862277030340796421, −5.01786326332934573911360804580, −4.26606229740453695291208601117, −3.17613585520047066847966027986, −2.40910225422251583507326352964, −1.99427337453769251688043633450, −0.63596818524233915662653421223, 0.63596818524233915662653421223, 1.99427337453769251688043633450, 2.40910225422251583507326352964, 3.17613585520047066847966027986, 4.26606229740453695291208601117, 5.01786326332934573911360804580, 5.84907127756862277030340796421, 6.52140951385644753733343816993, 7.31419616245497645615733726648, 7.903712515451102369404979111736

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