Properties

Label 2-8470-1.1-c1-0-28
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 + 1.16·3-s + 4-s − 5-s − 1.16·6-s + 7-s − 8-s − 1.63·9-s + 10-s + 1.16·12-s + 1.43·13-s − 14-s − 1.16·15-s + 16-s − 7.28·17-s + 1.63·18-s − 4.33·19-s − 20-s + 1.16·21-s + 2.02·23-s − 1.16·24-s + 25-s − 1.43·26-s − 5.41·27-s + 28-s + 9.11·29-s + 1.16·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.675·3-s + 0.5·4-s − 0.447·5-s − 0.477·6-s + 0.377·7-s − 0.353·8-s − 0.543·9-s + 0.316·10-s + 0.337·12-s + 0.397·13-s − 0.267·14-s − 0.302·15-s + 0.250·16-s − 1.76·17-s + 0.384·18-s − 0.994·19-s − 0.223·20-s + 0.255·21-s + 0.422·23-s − 0.238·24-s + 0.200·25-s − 0.281·26-s − 1.04·27-s + 0.188·28-s + 1.69·29-s + 0.213·30-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

\(L(1)\) \(\approx\) \(1.273842725\)
\(L(\frac12)\) \(\approx\) \(1.273842725\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.16T + 3T^{2} \)
13 \( 1 - 1.43T + 13T^{2} \)
17 \( 1 + 7.28T + 17T^{2} \)
19 \( 1 + 4.33T + 19T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 - 9.11T + 29T^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 + 3.11T + 37T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 + 0.705T + 43T^{2} \)
47 \( 1 - 4.00T + 47T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 6.62T + 61T^{2} \)
67 \( 1 + 1.89T + 67T^{2} \)
71 \( 1 - 0.421T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 + 3.98T + 79T^{2} \)
83 \( 1 - 7.15T + 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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

−8.091297961171911561894317380738, −7.18718295272664609771388731496, −6.59558213675607506064534296886, −5.96038416554034779006494068792, −4.82422799870668552835467923649, −4.25139118355938388080449936369, −3.27532200538429636127554623235, −2.54446588906533650367396748174, −1.85108613436538160961602832999, −0.57424432940998752590300254526, 0.57424432940998752590300254526, 1.85108613436538160961602832999, 2.54446588906533650367396748174, 3.27532200538429636127554623235, 4.25139118355938388080449936369, 4.82422799870668552835467923649, 5.96038416554034779006494068792, 6.59558213675607506064534296886, 7.18718295272664609771388731496, 8.091297961171911561894317380738

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