Properties

Label 2-8470-1.1-c1-0-21
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 + 0.529·3-s + 4-s + 5-s − 0.529·6-s − 7-s − 8-s − 2.71·9-s − 10-s + 0.529·12-s + 4.81·13-s + 14-s + 0.529·15-s + 16-s − 6.02·17-s + 2.71·18-s − 3.59·19-s + 20-s − 0.529·21-s − 7.06·23-s − 0.529·24-s + 25-s − 4.81·26-s − 3.02·27-s − 28-s − 0.353·29-s − 0.529·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.447·5-s − 0.216·6-s − 0.377·7-s − 0.353·8-s − 0.906·9-s − 0.316·10-s + 0.152·12-s + 1.33·13-s + 0.267·14-s + 0.136·15-s + 0.250·16-s − 1.46·17-s + 0.641·18-s − 0.824·19-s + 0.223·20-s − 0.115·21-s − 1.47·23-s − 0.108·24-s + 0.200·25-s − 0.943·26-s − 0.582·27-s − 0.188·28-s − 0.0656·29-s − 0.0966·30-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

\(L(1)\) \(\approx\) \(1.108095798\)
\(L(\frac12)\) \(\approx\) \(1.108095798\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.529T + 3T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
23 \( 1 + 7.06T + 23T^{2} \)
29 \( 1 + 0.353T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 - 1.64T + 37T^{2} \)
41 \( 1 + 9.35T + 41T^{2} \)
43 \( 1 + 9.13T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 4.64T + 53T^{2} \)
59 \( 1 - 2.78T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 9.43T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 + 8.04T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−8.145067399759047158875362417424, −7.02311330424738804953111563470, −6.41880582469074617588967206545, −5.99555074940147339503168688882, −5.18249164976202050948969985340, −3.99426194356017041608754436900, −3.46333657825402104351020432661, −2.31718787796917458322040865931, −1.95686254159834108161214656840, −0.53583477324452634195610478348, 0.53583477324452634195610478348, 1.95686254159834108161214656840, 2.31718787796917458322040865931, 3.46333657825402104351020432661, 3.99426194356017041608754436900, 5.18249164976202050948969985340, 5.99555074940147339503168688882, 6.41880582469074617588967206545, 7.02311330424738804953111563470, 8.145067399759047158875362417424

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