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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s + 11-s − 6·13-s − 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s − 2·29-s + 8·31-s − 33-s + 6·37-s + 6·39-s − 10·41-s + 4·43-s + 2·45-s − 8·47-s + 2·51-s + 6·53-s + 2·55-s − 4·57-s + 4·59-s + 10·61-s − 12·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.174·33-s + 0.986·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s + 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.529·57-s + 0.520·59-s + 1.28·61-s − 1.48·65-s + 1.46·67-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  =  0
Selberg data  =  $(2,\ 25872,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.760676022$
$L(\frac12)$  $\approx$  $1.760676022$
$L(\frac{3}{2})$   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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.24737776059849, −14.85630108034320, −14.10668473815935, −13.83449799743045, −13.09280185443335, −12.71553696526476, −12.00064788629764, −11.56826427753578, −11.17070040339216, −10.09368211115660, −9.921346462065168, −9.648621324712400, −8.771289287190100, −8.143156274612701, −7.348109614582402, −6.922258437979171, −6.318929572259154, −5.650607789298156, −5.138027725780116, −4.620980855625013, −3.855954065685880, −2.846448569937217, −2.291953454104632, −1.505886846315364, −0.5426587396491301, 0.5426587396491301, 1.505886846315364, 2.291953454104632, 2.846448569937217, 3.855954065685880, 4.620980855625013, 5.138027725780116, 5.650607789298156, 6.318929572259154, 6.922258437979171, 7.348109614582402, 8.143156274612701, 8.771289287190100, 9.648621324712400, 9.921346462065168, 10.09368211115660, 11.17070040339216, 11.56826427753578, 12.00064788629764, 12.71553696526476, 13.09280185443335, 13.83449799743045, 14.10668473815935, 14.85630108034320, 15.24737776059849

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