Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s − 11-s + 6·13-s − 6·17-s + 8·19-s − 25-s + 6·29-s − 8·35-s + 6·37-s + 10·41-s + 8·43-s + 9·49-s − 6·53-s − 2·55-s + 4·59-s − 2·61-s + 12·65-s + 12·67-s − 8·71-s + 2·73-s + 4·77-s + 4·79-s − 12·83-s − 12·85-s + 6·89-s − 24·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s − 0.301·11-s + 1.66·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.35·35-s + 0.986·37-s + 1.56·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s − 1.30·85-s + 0.635·89-s − 2.51·91-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  :  $\chi_{1584} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1584,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.776127903\)
\(L(\frac12)\)  \(\approx\)  \(1.776127903\)
\(L(\frac{3}{2})\)   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.402612326378998416643459368370, −8.919906421356101286450384033508, −7.81845995144891541405433174054, −6.76153137764114692580153019196, −6.15707753854340364126761944921, −5.62329098122753327257506148467, −4.29789560350206921563462558010, −3.29333770957049475081943753861, −2.45750776683153373018171759402, −0.959729037876911481426237589919, 0.959729037876911481426237589919, 2.45750776683153373018171759402, 3.29333770957049475081943753861, 4.29789560350206921563462558010, 5.62329098122753327257506148467, 6.15707753854340364126761944921, 6.76153137764114692580153019196, 7.81845995144891541405433174054, 8.919906421356101286450384033508, 9.402612326378998416643459368370

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