Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s − 11-s + 2·13-s − 2·15-s + 2·17-s − 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 33-s + 6·37-s − 2·39-s + 2·41-s + 2·45-s + 8·47-s − 2·51-s + 6·53-s − 2·55-s − 4·59-s − 6·61-s + 4·65-s + 4·67-s + 8·69-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s + 0.485·17-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.298·45-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.269·55-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.963·69-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25872\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{25872} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 25872,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−15.66073130420254, −15.07583274941111, −14.47599869375186, −13.85450788067837, −13.53337335435094, −12.88458346292339, −12.41426328643923, −11.85368958804360, −11.16881322652983, −10.74330814120970, −10.17730850756304, −9.594719803901925, −9.229782845816903, −8.447357411626412, −7.621431993020351, −7.428407664778823, −6.221112120336446, −6.143720069108650, −5.490964192257805, −4.979155947195316, −3.932626021379084, −3.656562947209835, −2.404262425671094, −1.955750998329035, −1.076442371788735, 0, 1.076442371788735, 1.955750998329035, 2.404262425671094, 3.656562947209835, 3.932626021379084, 4.979155947195316, 5.490964192257805, 6.143720069108650, 6.221112120336446, 7.428407664778823, 7.621431993020351, 8.447357411626412, 9.229782845816903, 9.594719803901925, 10.17730850756304, 10.74330814120970, 11.16881322652983, 11.85368958804360, 12.41426328643923, 12.88458346292339, 13.53337335435094, 13.85450788067837, 14.47599869375186, 15.07583274941111, 15.66073130420254

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