Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 \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  − 2·5-s − 7-s + 11-s − 6·13-s + 2·17-s + 8·19-s − 4·23-s − 25-s − 2·29-s + 8·31-s + 2·35-s + 6·37-s + 2·41-s − 8·43-s − 4·47-s + 49-s − 2·53-s − 2·55-s − 12·59-s + 10·61-s + 12·65-s + 12·67-s − 12·71-s + 10·73-s − 77-s + 8·79-s + 12·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 0.301·11-s − 1.66·13-s + 0.485·17-s + 1.83·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.312·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s − 0.269·55-s − 1.56·59-s + 1.28·61-s + 1.48·65-s + 1.46·67-s − 1.42·71-s + 1.17·73-s − 0.113·77-s + 0.900·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11088\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11088} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 11088,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 + T \)
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 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.64985411913639, −16.19605731513380, −15.69045480254107, −15.04954192877420, −14.61958100817109, −13.87884847298560, −13.50711757795424, −12.37352801822218, −12.30404810523695, −11.61810517486781, −11.25732633219826, −10.14433104743991, −9.713918629625938, −9.447660287350827, −8.271872098947002, −7.849984647159836, −7.351829609423269, −6.696834031723473, −5.881270992720608, −5.093098917038192, −4.534404372649437, −3.667225229487809, −3.088789050785461, −2.251564091360853, −1.023125066945223, 0, 1.023125066945223, 2.251564091360853, 3.088789050785461, 3.667225229487809, 4.534404372649437, 5.093098917038192, 5.881270992720608, 6.696834031723473, 7.351829609423269, 7.849984647159836, 8.271872098947002, 9.447660287350827, 9.713918629625938, 10.14433104743991, 11.25732633219826, 11.61810517486781, 12.30404810523695, 12.37352801822218, 13.50711757795424, 13.87884847298560, 14.61958100817109, 15.04954192877420, 15.69045480254107, 16.19605731513380, 16.64985411913639

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