Properties

Label 2-8470-1.1-c1-0-22
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 − 2.67·3-s + 4-s − 5-s − 2.67·6-s − 7-s + 8-s + 4.15·9-s − 10-s − 2.67·12-s + 3.11·13-s − 14-s + 2.67·15-s + 16-s + 0.564·17-s + 4.15·18-s − 7.52·19-s − 20-s + 2.67·21-s + 4.63·23-s − 2.67·24-s + 25-s + 3.11·26-s − 3.09·27-s − 28-s − 6.24·29-s + 2.67·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.54·3-s + 0.5·4-s − 0.447·5-s − 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.38·9-s − 0.316·10-s − 0.772·12-s + 0.864·13-s − 0.267·14-s + 0.690·15-s + 0.250·16-s + 0.136·17-s + 0.980·18-s − 1.72·19-s − 0.223·20-s + 0.583·21-s + 0.966·23-s − 0.546·24-s + 0.200·25-s + 0.611·26-s − 0.596·27-s − 0.188·28-s − 1.15·29-s + 0.488·30-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

\(L(1)\) \(\approx\) \(1.275216196\)
\(L(\frac12)\) \(\approx\) \(1.275216196\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.67T + 3T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 - 0.564T + 17T^{2} \)
19 \( 1 + 7.52T + 19T^{2} \)
23 \( 1 - 4.63T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 + 0.240T + 37T^{2} \)
41 \( 1 - 2.28T + 41T^{2} \)
43 \( 1 - 3.78T + 43T^{2} \)
47 \( 1 + 0.981T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 8.79T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 6.80T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 4.33T + 89T^{2} \)
97 \( 1 + 9.05T + 97T^{2} \)
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

−7.46062929677355646876561250198, −6.81517478865952322084505093756, −6.23658945777095304082561145657, −5.81094893843958128397835831951, −5.03969160221207235576394661061, −4.34505833478909161130487388078, −3.79591554086456601514667164294, −2.81038408745714698031185207386, −1.61997039094973048434086533981, −0.54623396562703542390174169692, 0.54623396562703542390174169692, 1.61997039094973048434086533981, 2.81038408745714698031185207386, 3.79591554086456601514667164294, 4.34505833478909161130487388078, 5.03969160221207235576394661061, 5.81094893843958128397835831951, 6.23658945777095304082561145657, 6.81517478865952322084505093756, 7.46062929677355646876561250198

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