Properties

Label 2-1584-1.1-c1-0-18
Degree $2$
Conductor $1584$
Sign $-1$
Analytic cond. $12.6483$
Root an. cond. $3.55644$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 11-s − 4·13-s + 6·17-s + 4·19-s + 6·23-s − 5·25-s − 6·29-s − 8·31-s − 10·37-s − 6·41-s − 8·43-s − 6·47-s − 3·49-s + 8·61-s + 4·67-s + 6·71-s + 2·73-s + 2·77-s − 14·79-s − 12·83-s + 6·89-s + 8·91-s + 14·97-s − 6·101-s + 4·103-s − 12·107-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 1.02·61-s + 0.488·67-s + 0.712·71-s + 0.234·73-s + 0.227·77-s − 1.57·79-s − 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s + 0.394·103-s − 1.16·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.245228842316309160133095693960, −8.129852226719742477393784673308, −7.33998580522115612678977642043, −6.79963832425739493066244038791, −5.43952609375822489573657963774, −5.21184114914749280771382385734, −3.64282367595579751815074401037, −3.08131485219503756183708188012, −1.70794665150414314375514308074, 0, 1.70794665150414314375514308074, 3.08131485219503756183708188012, 3.64282367595579751815074401037, 5.21184114914749280771382385734, 5.43952609375822489573657963774, 6.79963832425739493066244038791, 7.33998580522115612678977642043, 8.129852226719742477393784673308, 9.245228842316309160133095693960

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