Properties

Label 2-8470-1.1-c1-0-35
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 − 0.406·3-s + 4-s − 5-s − 0.406·6-s + 7-s + 8-s − 2.83·9-s − 10-s − 0.406·12-s + 0.813·13-s + 14-s + 0.406·15-s + 16-s − 3.83·17-s − 2.83·18-s − 5.42·19-s − 20-s − 0.406·21-s − 5.42·23-s − 0.406·24-s + 25-s + 0.813·26-s + 2.37·27-s + 28-s − 6.61·29-s + 0.406·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.234·3-s + 0.5·4-s − 0.447·5-s − 0.166·6-s + 0.377·7-s + 0.353·8-s − 0.944·9-s − 0.316·10-s − 0.117·12-s + 0.225·13-s + 0.267·14-s + 0.105·15-s + 0.250·16-s − 0.930·17-s − 0.668·18-s − 1.24·19-s − 0.223·20-s − 0.0887·21-s − 1.13·23-s − 0.0830·24-s + 0.200·25-s + 0.159·26-s + 0.456·27-s + 0.188·28-s − 1.22·29-s + 0.0742·30-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

\(L(1)\) \(\approx\) \(2.077886330\)
\(L(\frac12)\) \(\approx\) \(2.077886330\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.406T + 3T^{2} \)
13 \( 1 - 0.813T + 13T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 + 5.42T + 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 5.59T + 41T^{2} \)
43 \( 1 - 3.02T + 43T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 0.978T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 6.20T + 73T^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 - 8.81T + 83T^{2} \)
89 \( 1 + 1.18T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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

−7.88268991673752088675263470385, −6.91320434864749270704982969055, −6.25822424405428322403524416026, −5.77878172686810871034776302648, −4.93908708596876869749480126943, −4.21755423131753341217116588301, −3.74106618821959260752382278174, −2.58612239454549606812151712882, −2.08269360183135597600494968686, −0.61200122363716809953341445245, 0.61200122363716809953341445245, 2.08269360183135597600494968686, 2.58612239454549606812151712882, 3.74106618821959260752382278174, 4.21755423131753341217116588301, 4.93908708596876869749480126943, 5.77878172686810871034776302648, 6.25822424405428322403524416026, 6.91320434864749270704982969055, 7.88268991673752088675263470385

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