Properties

Label 2-8470-1.1-c1-0-157
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 − 2.54·3-s + 4-s + 5-s − 2.54·6-s + 7-s + 8-s + 3.46·9-s + 10-s − 2.54·12-s − 6.42·13-s + 14-s − 2.54·15-s + 16-s − 3.95·17-s + 3.46·18-s + 2.46·19-s + 20-s − 2.54·21-s + 4.99·23-s − 2.54·24-s + 25-s − 6.42·26-s − 1.19·27-s + 28-s − 9.04·29-s − 2.54·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.46·3-s + 0.5·4-s + 0.447·5-s − 1.03·6-s + 0.377·7-s + 0.353·8-s + 1.15·9-s + 0.316·10-s − 0.734·12-s − 1.78·13-s + 0.267·14-s − 0.656·15-s + 0.250·16-s − 0.958·17-s + 0.817·18-s + 0.566·19-s + 0.223·20-s − 0.555·21-s + 1.04·23-s − 0.519·24-s + 0.200·25-s − 1.25·26-s − 0.229·27-s + 0.188·28-s − 1.67·29-s − 0.464·30-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: \(1\)
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 \( 1 + 2.54T + 3T^{2} \)
13 \( 1 + 6.42T + 13T^{2} \)
17 \( 1 + 3.95T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 + 9.04T + 29T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 6.95T + 41T^{2} \)
43 \( 1 - 3.02T + 43T^{2} \)
47 \( 1 + 8.37T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 3.59T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 5.90T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.16108861985579202703026156878, −6.64486338621118466527963907519, −5.86964229828567738771834607749, −5.36588050435770427081491629219, −4.70044080302775976862849784172, −4.40141239893183279096390549541, −3.01040462017020156443432917075, −2.26913551880451943458853095703, −1.21652336148100754141193278792, 0, 1.21652336148100754141193278792, 2.26913551880451943458853095703, 3.01040462017020156443432917075, 4.40141239893183279096390549541, 4.70044080302775976862849784172, 5.36588050435770427081491629219, 5.86964229828567738771834607749, 6.64486338621118466527963907519, 7.16108861985579202703026156878

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