Properties

Label 2-8470-1.1-c1-0-140
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.82·3-s + 4-s − 5-s − 1.82·6-s + 7-s + 8-s + 0.328·9-s − 10-s − 1.82·12-s − 2.21·13-s + 14-s + 1.82·15-s + 16-s − 7.96·17-s + 0.328·18-s − 0.915·19-s − 20-s − 1.82·21-s + 6.07·23-s − 1.82·24-s + 25-s − 2.21·26-s + 4.87·27-s + 28-s + 5.06·29-s + 1.82·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.05·3-s + 0.5·4-s − 0.447·5-s − 0.744·6-s + 0.377·7-s + 0.353·8-s + 0.109·9-s − 0.316·10-s − 0.526·12-s − 0.614·13-s + 0.267·14-s + 0.471·15-s + 0.250·16-s − 1.93·17-s + 0.0773·18-s − 0.210·19-s − 0.223·20-s − 0.398·21-s + 1.26·23-s − 0.372·24-s + 0.200·25-s − 0.434·26-s + 0.938·27-s + 0.188·28-s + 0.940·29-s + 0.333·30-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 \( 1 + 1.82T + 3T^{2} \)
13 \( 1 + 2.21T + 13T^{2} \)
17 \( 1 + 7.96T + 17T^{2} \)
19 \( 1 + 0.915T + 19T^{2} \)
23 \( 1 - 6.07T + 23T^{2} \)
29 \( 1 - 5.06T + 29T^{2} \)
31 \( 1 - 9.33T + 31T^{2} \)
37 \( 1 - 4.32T + 37T^{2} \)
41 \( 1 + 8.99T + 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 4.22T + 47T^{2} \)
53 \( 1 + 9.33T + 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 1.29T + 67T^{2} \)
71 \( 1 - 0.0202T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 9.39T + 79T^{2} \)
83 \( 1 - 0.747T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 7.27T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.03441453839236369406788921620, −6.70856425075925496535603765500, −6.07844326298878534312235074087, −5.18966279768516864879567378695, −4.62900168774724117475754131708, −4.33124130547277825933337546245, −3.02602786217034420862124818489, −2.42557439155520253820831286658, −1.14357868511710481442022382815, 0, 1.14357868511710481442022382815, 2.42557439155520253820831286658, 3.02602786217034420862124818489, 4.33124130547277825933337546245, 4.62900168774724117475754131708, 5.18966279768516864879567378695, 6.07844326298878534312235074087, 6.70856425075925496535603765500, 7.03441453839236369406788921620

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