Properties

Label 2-8470-1.1-c1-0-174
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 − 1.32·3-s + 4-s + 5-s − 1.32·6-s − 7-s + 8-s − 1.25·9-s + 10-s − 1.32·12-s + 5.34·13-s − 14-s − 1.32·15-s + 16-s − 6.44·17-s − 1.25·18-s − 6.42·19-s + 20-s + 1.32·21-s + 5.51·23-s − 1.32·24-s + 25-s + 5.34·26-s + 5.62·27-s − 28-s − 8.19·29-s − 1.32·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.762·3-s + 0.5·4-s + 0.447·5-s − 0.539·6-s − 0.377·7-s + 0.353·8-s − 0.418·9-s + 0.316·10-s − 0.381·12-s + 1.48·13-s − 0.267·14-s − 0.341·15-s + 0.250·16-s − 1.56·17-s − 0.295·18-s − 1.47·19-s + 0.223·20-s + 0.288·21-s + 1.14·23-s − 0.269·24-s + 0.200·25-s + 1.04·26-s + 1.08·27-s − 0.188·28-s − 1.52·29-s − 0.241·30-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 \( 1 + 1.32T + 3T^{2} \)
13 \( 1 - 5.34T + 13T^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + 6.42T + 19T^{2} \)
23 \( 1 - 5.51T + 23T^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 - 1.77T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 2.08T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 - 4.35T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 + 5.44T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 5.54T + 73T^{2} \)
79 \( 1 + 7.35T + 79T^{2} \)
83 \( 1 + 7.19T + 83T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + 0.548T + 97T^{2} \)
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

−7.06267565364705965642747032621, −6.46406774915150690874162506571, −6.04581469702431695364915540155, −5.56845076375964506269095173617, −4.56097764827422491705912452208, −4.12263118232089893232358714677, −3.08033679931687761788169513508, −2.35137843060326188664069937509, −1.30973733228809207115518102709, 0, 1.30973733228809207115518102709, 2.35137843060326188664069937509, 3.08033679931687761788169513508, 4.12263118232089893232358714677, 4.56097764827422491705912452208, 5.56845076375964506269095173617, 6.04581469702431695364915540155, 6.46406774915150690874162506571, 7.06267565364705965642747032621

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