Properties

Label 2-8470-1.1-c1-0-181
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 + 2.61·3-s + 4-s − 5-s − 2.61·6-s − 7-s − 8-s + 3.85·9-s + 10-s + 2.61·12-s − 2·13-s + 14-s − 2.61·15-s + 16-s + 1.61·17-s − 3.85·18-s − 6.85·19-s − 20-s − 2.61·21-s + 6·23-s − 2.61·24-s + 25-s + 2·26-s + 2.23·27-s − 28-s − 3.23·29-s + 2.61·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.447·5-s − 1.06·6-s − 0.377·7-s − 0.353·8-s + 1.28·9-s + 0.316·10-s + 0.755·12-s − 0.554·13-s + 0.267·14-s − 0.675·15-s + 0.250·16-s + 0.392·17-s − 0.908·18-s − 1.57·19-s − 0.223·20-s − 0.571·21-s + 1.25·23-s − 0.534·24-s + 0.200·25-s + 0.392·26-s + 0.430·27-s − 0.188·28-s − 0.600·29-s + 0.477·30-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: \(1\)
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 \( 1 - 2.61T + 3T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + 6.85T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 3.23T + 29T^{2} \)
31 \( 1 + 1.23T + 31T^{2} \)
37 \( 1 + 6.47T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 - 9.23T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 + 7.61T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + 9.70T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 8.47T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 9.56T + 97T^{2} \)
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

−7.46264429169674004532743692244, −7.21189559697520536471377097697, −6.34922902067241816381709942199, −5.38928741433899459529737440625, −4.30819419361542607104746686627, −3.71539330290363594427277486396, −2.84849362557972434686489115053, −2.38245051820441006457868111340, −1.37022022837728817710405307204, 0, 1.37022022837728817710405307204, 2.38245051820441006457868111340, 2.84849362557972434686489115053, 3.71539330290363594427277486396, 4.30819419361542607104746686627, 5.38928741433899459529737440625, 6.34922902067241816381709942199, 7.21189559697520536471377097697, 7.46264429169674004532743692244

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