Properties

Label 2-585-1.1-c1-0-13
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + 2.41·2-s + 3.82·4-s − 5-s + 4.82·7-s + 4.41·8-s − 2.41·10-s − 3.41·11-s − 13-s + 11.6·14-s + 2.99·16-s − 0.828·17-s + 0.585·19-s − 3.82·20-s − 8.24·22-s − 1.41·23-s + 25-s − 2.41·26-s + 18.4·28-s + 5.65·29-s + 1.75·31-s − 1.58·32-s − 1.99·34-s − 4.82·35-s − 8.48·37-s + 1.41·38-s − 4.41·40-s + 3.17·41-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s − 0.447·5-s + 1.82·7-s + 1.56·8-s − 0.763·10-s − 1.02·11-s − 0.277·13-s + 3.11·14-s + 0.749·16-s − 0.200·17-s + 0.134·19-s − 0.856·20-s − 1.75·22-s − 0.294·23-s + 0.200·25-s − 0.473·26-s + 3.49·28-s + 1.05·29-s + 0.315·31-s − 0.280·32-s − 0.342·34-s − 0.816·35-s − 1.39·37-s + 0.229·38-s − 0.697·40-s + 0.495·41-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: no
Arithmetic: yes
Character: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.876629761\)
\(L(\frac12)\) \(\approx\) \(3.876629761\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 3.17T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 + 2.48T + 53T^{2} \)
59 \( 1 + 1.75T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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

−11.02327904860097487853365600695, −10.36110934961180151067116332977, −8.601144137217098123137355612783, −7.82748707143792572786159821091, −7.00274043662204370004099982388, −5.70839080855422604232435173677, −4.89361651758623846143271627664, −4.42392998264889679461814329749, −3.07921870702749460485365586304, −1.91204956050814003374058696388, 1.91204956050814003374058696388, 3.07921870702749460485365586304, 4.42392998264889679461814329749, 4.89361651758623846143271627664, 5.70839080855422604232435173677, 7.00274043662204370004099982388, 7.82748707143792572786159821091, 8.601144137217098123137355612783, 10.36110934961180151067116332977, 11.02327904860097487853365600695

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