Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

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 + 101-s + 103-s + 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 + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\( d \)  =  \(2\)
\( N \)  =  \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1584} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1584,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 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 \( 1 - 14 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.55014512936591, −18.79419118382526, −18.47747140403480, −17.36562787398367, −16.91921474016255, −16.25020643671804, −15.58225952630799, −14.76507429446245, −14.28260474550574, −13.28226608919633, −12.81798063706554, −12.01488628985350, −11.42409110725976, −10.34192128775270, −9.779706047686582, −9.245228842316309, −8.129852226719742, −7.339985805221156, −6.799638324257395, −5.439526093758225, −5.211841149147493, −3.642823675955798, −3.081314852195038, −1.707946651504143, 0, 1.707946651504143, 3.081314852195038, 3.642823675955798, 5.211841149147493, 5.439526093758225, 6.799638324257395, 7.339985805221156, 8.129852226719742, 9.245228842316309, 9.779706047686582, 10.34192128775270, 11.42409110725976, 12.01488628985350, 12.81798063706554, 13.28226608919633, 14.28260474550574, 14.76507429446245, 15.58225952630799, 16.25020643671804, 16.91921474016255, 17.36562787398367, 18.47747140403480, 18.79419118382526, 19.55014512936591

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