Properties

Label 2-567-1.1-c3-0-22
Degree $2$
Conductor $567$
Sign $1$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.807·2-s − 7.34·4-s + 18.2·5-s − 7·7-s − 12.3·8-s + 14.7·10-s − 49.4·11-s + 44.4·13-s − 5.65·14-s + 48.7·16-s + 47.2·17-s + 56.7·19-s − 133.·20-s − 39.9·22-s − 55.4·23-s + 207.·25-s + 35.9·26-s + 51.4·28-s + 150.·29-s − 167.·31-s + 138.·32-s + 38.1·34-s − 127.·35-s + 331.·37-s + 45.8·38-s − 225.·40-s + 295.·41-s + ⋯
L(s)  = 1  + 0.285·2-s − 0.918·4-s + 1.63·5-s − 0.377·7-s − 0.547·8-s + 0.465·10-s − 1.35·11-s + 0.949·13-s − 0.107·14-s + 0.762·16-s + 0.673·17-s + 0.685·19-s − 1.49·20-s − 0.386·22-s − 0.502·23-s + 1.65·25-s + 0.270·26-s + 0.347·28-s + 0.962·29-s − 0.970·31-s + 0.765·32-s + 0.192·34-s − 0.616·35-s + 1.47·37-s + 0.195·38-s − 0.892·40-s + 1.12·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.291951215\)
\(L(\frac12)\) \(\approx\) \(2.291951215\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 - 0.807T + 8T^{2} \)
5 \( 1 - 18.2T + 125T^{2} \)
11 \( 1 + 49.4T + 1.33e3T^{2} \)
13 \( 1 - 44.4T + 2.19e3T^{2} \)
17 \( 1 - 47.2T + 4.91e3T^{2} \)
19 \( 1 - 56.7T + 6.85e3T^{2} \)
23 \( 1 + 55.4T + 1.21e4T^{2} \)
29 \( 1 - 150.T + 2.43e4T^{2} \)
31 \( 1 + 167.T + 2.97e4T^{2} \)
37 \( 1 - 331.T + 5.06e4T^{2} \)
41 \( 1 - 295.T + 6.89e4T^{2} \)
43 \( 1 + 317.T + 7.95e4T^{2} \)
47 \( 1 + 137.T + 1.03e5T^{2} \)
53 \( 1 - 411.T + 1.48e5T^{2} \)
59 \( 1 - 212.T + 2.05e5T^{2} \)
61 \( 1 + 52.3T + 2.26e5T^{2} \)
67 \( 1 - 706.T + 3.00e5T^{2} \)
71 \( 1 + 78.7T + 3.57e5T^{2} \)
73 \( 1 - 839.T + 3.89e5T^{2} \)
79 \( 1 - 1.01e3T + 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + 762.T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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

−10.01112299547976561420205120756, −9.702268294250300235283971760278, −8.718437056256337295642065422119, −7.80739951212169852179757687182, −6.32564299820104854278790889700, −5.63565389972906301315918629928, −5.00118678792501349202734527756, −3.54521156394464639328599815003, −2.43796025571670008641189145756, −0.906559265284840969815175048477, 0.906559265284840969815175048477, 2.43796025571670008641189145756, 3.54521156394464639328599815003, 5.00118678792501349202734527756, 5.63565389972906301315918629928, 6.32564299820104854278790889700, 7.80739951212169852179757687182, 8.718437056256337295642065422119, 9.702268294250300235283971760278, 10.01112299547976561420205120756

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