Properties

Label 2-8470-1.1-c1-0-132
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 + 2.61·3-s + 4-s − 5-s + 2.61·6-s + 7-s + 8-s + 3.85·9-s − 10-s + 2.61·12-s + 2·13-s + 14-s − 2.61·15-s + 16-s − 1.61·17-s + 3.85·18-s + 6.85·19-s − 20-s + 2.61·21-s + 6·23-s + 2.61·24-s + 25-s + 2·26-s + 2.23·27-s + 28-s + 3.23·29-s − 2.61·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.447·5-s + 1.06·6-s + 0.377·7-s + 0.353·8-s + 1.28·9-s − 0.316·10-s + 0.755·12-s + 0.554·13-s + 0.267·14-s − 0.675·15-s + 0.250·16-s − 0.392·17-s + 0.908·18-s + 1.57·19-s − 0.223·20-s + 0.571·21-s + 1.25·23-s + 0.534·24-s + 0.200·25-s + 0.392·26-s + 0.430·27-s + 0.188·28-s + 0.600·29-s − 0.477·30-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

\(L(1)\) \(\approx\) \(6.385163308\)
\(L(\frac12)\) \(\approx\) \(6.385163308\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 3T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 - 6.85T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 + 1.23T + 31T^{2} \)
37 \( 1 + 6.47T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 - 9.23T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 + 7.61T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + 9.70T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 8.47T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 9.56T + 97T^{2} \)
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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

−7.74515238481860124047935888291, −7.22583259874961772267331316318, −6.61251981423924699059184993406, −5.49354971531854480003293368294, −4.88775267676840082692348819403, −4.06518134710107328686888570203, −3.34833684309992433015003167530, −2.97733729146167180574103667338, −2.00348214207417670392999231143, −1.11082580664517403773910388091, 1.11082580664517403773910388091, 2.00348214207417670392999231143, 2.97733729146167180574103667338, 3.34833684309992433015003167530, 4.06518134710107328686888570203, 4.88775267676840082692348819403, 5.49354971531854480003293368294, 6.61251981423924699059184993406, 7.22583259874961772267331316318, 7.74515238481860124047935888291

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