Properties

Label 2-585-1.1-c1-0-3
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 7-s + 3·11-s + 13-s + 4·16-s + 3·17-s − 4·19-s + 2·20-s + 9·23-s + 25-s + 2·28-s + 6·29-s + 2·31-s + 35-s − 37-s + 3·41-s + 2·43-s − 6·44-s + 6·47-s − 6·49-s − 2·52-s − 9·53-s − 3·55-s + 12·59-s + 5·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 0.377·7-s + 0.904·11-s + 0.277·13-s + 16-s + 0.727·17-s − 0.917·19-s + 0.447·20-s + 1.87·23-s + 1/5·25-s + 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.169·35-s − 0.164·37-s + 0.468·41-s + 0.304·43-s − 0.904·44-s + 0.875·47-s − 6/7·49-s − 0.277·52-s − 1.23·53-s − 0.404·55-s + 1.56·59-s + 0.640·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082399395\)
\(L(\frac12)\) \(\approx\) \(1.082399395\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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

−10.62510200039904674439234856658, −9.713611393939176921168495431920, −8.906263740943597793516749236173, −8.292416872807615906112471664419, −7.12261475792635686075443478244, −6.16181348679141975757729496775, −4.96599442706405448365610251513, −4.07858389118534067025312612810, −3.11322967677606911671732631614, −0.968476842474168100184639004266, 0.968476842474168100184639004266, 3.11322967677606911671732631614, 4.07858389118534067025312612810, 4.96599442706405448365610251513, 6.16181348679141975757729496775, 7.12261475792635686075443478244, 8.292416872807615906112471664419, 8.906263740943597793516749236173, 9.713611393939176921168495431920, 10.62510200039904674439234856658

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