Properties

Label 2-8470-1.1-c1-0-183
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 + 2.00·3-s + 4-s − 5-s − 2.00·6-s + 7-s − 8-s + 1.01·9-s + 10-s + 2.00·12-s + 4.02·13-s − 14-s − 2.00·15-s + 16-s − 0.300·17-s − 1.01·18-s − 7.73·19-s − 20-s + 2.00·21-s − 6.48·23-s − 2.00·24-s + 25-s − 4.02·26-s − 3.97·27-s + 28-s − 1.81·29-s + 2.00·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 0.5·4-s − 0.447·5-s − 0.818·6-s + 0.377·7-s − 0.353·8-s + 0.339·9-s + 0.316·10-s + 0.578·12-s + 1.11·13-s − 0.267·14-s − 0.517·15-s + 0.250·16-s − 0.0728·17-s − 0.239·18-s − 1.77·19-s − 0.223·20-s + 0.437·21-s − 1.35·23-s − 0.409·24-s + 0.200·25-s − 0.790·26-s − 0.764·27-s + 0.188·28-s − 0.337·29-s + 0.365·30-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 \( 1 - 2.00T + 3T^{2} \)
13 \( 1 - 4.02T + 13T^{2} \)
17 \( 1 + 0.300T + 17T^{2} \)
19 \( 1 + 7.73T + 19T^{2} \)
23 \( 1 + 6.48T + 23T^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 - 7.53T + 31T^{2} \)
37 \( 1 - 9.11T + 37T^{2} \)
41 \( 1 + 9.68T + 41T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 - 6.39T + 53T^{2} \)
59 \( 1 - 5.51T + 59T^{2} \)
61 \( 1 - 3.62T + 61T^{2} \)
67 \( 1 - 0.505T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 + 6.20T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 1.92T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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

−7.929197648090870461071986900963, −6.82733227781383901549810412023, −6.35121458077525786016015791999, −5.47741143046712229417022634395, −4.21852130259246353007590324593, −3.89994750658933222138591568328, −2.89317522561712016039164328629, −2.20329386492109827194972727999, −1.38870508801184222864486358109, 0, 1.38870508801184222864486358109, 2.20329386492109827194972727999, 2.89317522561712016039164328629, 3.89994750658933222138591568328, 4.21852130259246353007590324593, 5.47741143046712229417022634395, 6.35121458077525786016015791999, 6.82733227781383901549810412023, 7.929197648090870461071986900963

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