Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11^{2} $
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 − 2·7-s + 9-s + 2·13-s − 2·15-s − 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s + 8·29-s + 8·31-s − 4·35-s + 10·37-s − 2·39-s − 8·41-s − 2·43-s + 2·45-s + 8·47-s − 3·49-s + 4·51-s − 2·53-s + 6·57-s − 12·59-s − 10·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.304·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.794·57-s − 1.56·59-s − 1.28·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5808\)    =    \(2^{4} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5808} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5808,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−17.76277073243863, −17.16858596077503, −16.74009500545657, −16.03969385584500, −15.43702511922673, −14.99286700249704, −13.88696870912500, −13.57468343843999, −13.03049902647713, −12.38623518606749, −11.74061148913717, −10.96771890526442, −10.33253443605309, −9.977777335778390, −9.112867177724677, −8.610805139739024, −7.727791611228101, −6.669843389625779, −6.219388970049515, −6.001419282883136, −4.714837015043081, −4.344046137163675, −3.113368884044088, −2.325872817634532, −1.306074551377245, 0, 1.306074551377245, 2.325872817634532, 3.113368884044088, 4.344046137163675, 4.714837015043081, 6.001419282883136, 6.219388970049515, 6.669843389625779, 7.727791611228101, 8.610805139739024, 9.112867177724677, 9.977777335778390, 10.33253443605309, 10.96771890526442, 11.74061148913717, 12.38623518606749, 13.03049902647713, 13.57468343843999, 13.88696870912500, 14.99286700249704, 15.43702511922673, 16.03969385584500, 16.74009500545657, 17.16858596077503, 17.76277073243863

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