Properties

Label 2-1584-1.1-c3-0-8
Degree $2$
Conductor $1584$
Sign $1$
Analytic cond. $93.4590$
Root an. cond. $9.66742$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 10·5-s − 8·7-s − 11·11-s + 18·13-s − 46·17-s − 40·19-s + 44·23-s − 25·25-s − 186·29-s + 72·31-s + 80·35-s − 114·37-s − 174·41-s + 416·43-s − 156·47-s − 279·49-s + 62·53-s + 110·55-s − 348·59-s − 446·61-s − 180·65-s + 956·67-s − 444·71-s + 306·73-s + 88·77-s + 664·79-s − 124·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.431·7-s − 0.301·11-s + 0.384·13-s − 0.656·17-s − 0.482·19-s + 0.398·23-s − 1/5·25-s − 1.19·29-s + 0.417·31-s + 0.386·35-s − 0.506·37-s − 0.662·41-s + 1.47·43-s − 0.484·47-s − 0.813·49-s + 0.160·53-s + 0.269·55-s − 0.767·59-s − 0.936·61-s − 0.343·65-s + 1.74·67-s − 0.742·71-s + 0.490·73-s + 0.130·77-s + 0.945·79-s − 0.163·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(93.4590\)
Root analytic conductor: \(9.66742\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9654754689\)
\(L(\frac12)\) \(\approx\) \(0.9654754689\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + p T \)
good5 \( 1 + 2 p T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 - 18 T + p^{3} T^{2} \)
17 \( 1 + 46 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 - 44 T + p^{3} T^{2} \)
29 \( 1 + 186 T + p^{3} T^{2} \)
31 \( 1 - 72 T + p^{3} T^{2} \)
37 \( 1 + 114 T + p^{3} T^{2} \)
41 \( 1 + 174 T + p^{3} T^{2} \)
43 \( 1 - 416 T + p^{3} T^{2} \)
47 \( 1 + 156 T + p^{3} T^{2} \)
53 \( 1 - 62 T + p^{3} T^{2} \)
59 \( 1 + 348 T + p^{3} T^{2} \)
61 \( 1 + 446 T + p^{3} T^{2} \)
67 \( 1 - 956 T + p^{3} T^{2} \)
71 \( 1 + 444 T + p^{3} T^{2} \)
73 \( 1 - 306 T + p^{3} T^{2} \)
79 \( 1 - 664 T + p^{3} T^{2} \)
83 \( 1 + 124 T + p^{3} T^{2} \)
89 \( 1 + 602 T + p^{3} T^{2} \)
97 \( 1 - 1522 T + p^{3} T^{2} \)
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

−8.997280840593646172930709669131, −8.228494224478681748291433559640, −7.50670426278806939191199115363, −6.69827041725272217875261075640, −5.86431424814608006959016276003, −4.78868422281913966046189780531, −3.93859234780406837073225619374, −3.15320935663882896771997484711, −1.94383997361586669549720035334, −0.45876480818139869472459027168, 0.45876480818139869472459027168, 1.94383997361586669549720035334, 3.15320935663882896771997484711, 3.93859234780406837073225619374, 4.78868422281913966046189780531, 5.86431424814608006959016276003, 6.69827041725272217875261075640, 7.50670426278806939191199115363, 8.228494224478681748291433559640, 8.997280840593646172930709669131

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