Properties

Label 2-8470-1.1-c1-0-4
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.53·3-s + 4-s − 5-s − 2.53·6-s − 7-s + 8-s + 3.44·9-s − 10-s − 2.53·12-s − 4.11·13-s − 14-s + 2.53·15-s + 16-s − 6.84·17-s + 3.44·18-s − 6.62·19-s − 20-s + 2.53·21-s − 3.13·23-s − 2.53·24-s + 25-s − 4.11·26-s − 1.13·27-s − 28-s + 5.49·29-s + 2.53·30-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 + 1.14·9-s − 0.316·10-s − 0.733·12-s − 1.14·13-s − 0.267·14-s + 0.655·15-s + 0.250·16-s − 1.66·17-s + 0.812·18-s − 1.51·19-s − 0.223·20-s + 0.554·21-s − 0.654·23-s − 0.518·24-s + 0.200·25-s − 0.806·26-s − 0.219·27-s − 0.188·28-s + 1.02·29-s + 0.463·30-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

\(L(1)\) \(\approx\) \(0.3956396380\)
\(L(\frac12)\) \(\approx\) \(0.3956396380\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.53T + 3T^{2} \)
13 \( 1 + 4.11T + 13T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
19 \( 1 + 6.62T + 19T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 - 5.49T + 29T^{2} \)
31 \( 1 + 5.57T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 - 2.90T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + 1.49T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 7.42T + 59T^{2} \)
61 \( 1 + 2.28T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 0.0608T + 71T^{2} \)
73 \( 1 + 3.91T + 73T^{2} \)
79 \( 1 + 4.23T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 6.54T + 89T^{2} \)
97 \( 1 + 0.691T + 97T^{2} \)
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

−7.46300497608780960993241303012, −6.70052887534161112314846506227, −6.47772336946103016708727101341, −5.72441356458521956394307144254, −4.84566756246616418538584114006, −4.55213567574086037029855236028, −3.79155090573326289255349787236, −2.64921991915006968329801242433, −1.84342289878337857735896068073, −0.28039423432335704463269953369, 0.28039423432335704463269953369, 1.84342289878337857735896068073, 2.64921991915006968329801242433, 3.79155090573326289255349787236, 4.55213567574086037029855236028, 4.84566756246616418538584114006, 5.72441356458521956394307144254, 6.47772336946103016708727101341, 6.70052887534161112314846506227, 7.46300497608780960993241303012

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