Properties

Label 2-8470-1.1-c1-0-155
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 − 2.73·3-s + 4-s + 5-s − 2.73·6-s − 7-s + 8-s + 4.50·9-s + 10-s − 2.73·12-s + 2.65·13-s − 14-s − 2.73·15-s + 16-s − 6.67·17-s + 4.50·18-s + 5.92·19-s + 20-s + 2.73·21-s − 6.62·23-s − 2.73·24-s + 25-s + 2.65·26-s − 4.11·27-s − 28-s + 1.10·29-s − 2.73·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.447·5-s − 1.11·6-s − 0.377·7-s + 0.353·8-s + 1.50·9-s + 0.316·10-s − 0.790·12-s + 0.735·13-s − 0.267·14-s − 0.707·15-s + 0.250·16-s − 1.61·17-s + 1.06·18-s + 1.35·19-s + 0.223·20-s + 0.597·21-s − 1.38·23-s − 0.559·24-s + 0.200·25-s + 0.520·26-s − 0.792·27-s − 0.188·28-s + 0.205·29-s − 0.500·30-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: \(1\)
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 \( 1 + 2.73T + 3T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 + 6.67T + 17T^{2} \)
19 \( 1 - 5.92T + 19T^{2} \)
23 \( 1 + 6.62T + 23T^{2} \)
29 \( 1 - 1.10T + 29T^{2} \)
31 \( 1 + 5.57T + 31T^{2} \)
37 \( 1 - 4.15T + 37T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 - 0.907T + 47T^{2} \)
53 \( 1 + 9.24T + 53T^{2} \)
59 \( 1 - 1.72T + 59T^{2} \)
61 \( 1 + 8.93T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 - 0.967T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−7.13055712662144139364084128739, −6.34389471507509258150136341409, −6.09717559071919470446669505765, −5.49348592900271100448739389174, −4.72802075961333601766022001468, −4.15792849217948105169475698394, −3.23558749106915858518544225028, −2.14818008107017216321183840285, −1.20750080573714985619795575227, 0, 1.20750080573714985619795575227, 2.14818008107017216321183840285, 3.23558749106915858518544225028, 4.15792849217948105169475698394, 4.72802075961333601766022001468, 5.49348592900271100448739389174, 6.09717559071919470446669505765, 6.34389471507509258150136341409, 7.13055712662144139364084128739

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