Properties

Label 2-8470-1.1-c1-0-161
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 − 3.14·3-s + 4-s + 5-s − 3.14·6-s + 7-s + 8-s + 6.91·9-s + 10-s − 3.14·12-s − 5.43·13-s + 14-s − 3.14·15-s + 16-s + 0.477·17-s + 6.91·18-s + 5.91·19-s + 20-s − 3.14·21-s − 2.65·23-s − 3.14·24-s + 25-s − 5.43·26-s − 12.3·27-s + 28-s + 6.56·29-s − 3.14·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.447·5-s − 1.28·6-s + 0.377·7-s + 0.353·8-s + 2.30·9-s + 0.316·10-s − 0.909·12-s − 1.50·13-s + 0.267·14-s − 0.813·15-s + 0.250·16-s + 0.115·17-s + 1.63·18-s + 1.35·19-s + 0.223·20-s − 0.687·21-s − 0.553·23-s − 0.642·24-s + 0.200·25-s − 1.06·26-s − 2.37·27-s + 0.188·28-s + 1.21·29-s − 0.574·30-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 \( 1 + 3.14T + 3T^{2} \)
13 \( 1 + 5.43T + 13T^{2} \)
17 \( 1 - 0.477T + 17T^{2} \)
19 \( 1 - 5.91T + 19T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 + 9.68T + 31T^{2} \)
37 \( 1 + 4.56T + 37T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 9.42T + 47T^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 + 0.566T + 59T^{2} \)
61 \( 1 + 2.04T + 61T^{2} \)
67 \( 1 + 1.24T + 67T^{2} \)
71 \( 1 + 3.62T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 3.38T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 + 9.54T + 89T^{2} \)
97 \( 1 + 3.93T + 97T^{2} \)
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

−7.17000976915584794206936911604, −6.68099972255142647977653042162, −5.73412871584645390868738769951, −5.41471964014715489249757851130, −4.89094808876283128385410572679, −4.26137037331529387689004716571, −3.19425624484420317891396926597, −2.06798211995058781350165300317, −1.23465302267431994723338321951, 0, 1.23465302267431994723338321951, 2.06798211995058781350165300317, 3.19425624484420317891396926597, 4.26137037331529387689004716571, 4.89094808876283128385410572679, 5.41471964014715489249757851130, 5.73412871584645390868738769951, 6.68099972255142647977653042162, 7.17000976915584794206936911604

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