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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 11-s − 4·17-s − 4·19-s − 4·23-s − 25-s − 2·29-s + 2·31-s + 2·35-s − 6·37-s − 4·41-s + 4·43-s + 2·47-s + 49-s − 2·53-s + 2·55-s − 6·59-s + 4·61-s − 12·71-s + 16·73-s + 77-s + 8·79-s − 12·83-s + 8·85-s − 10·89-s + 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 0.301·11-s − 0.970·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.338·35-s − 0.986·37-s − 0.624·41-s + 0.609·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s + 0.269·55-s − 0.781·59-s + 0.512·61-s − 1.42·71-s + 1.87·73-s + 0.113·77-s + 0.900·79-s − 1.31·83-s + 0.867·85-s − 1.05·89-s + 0.820·95-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  =  \(0\)
Selberg data  =  \((2,\ 11088,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.5735399128\)
\(L(\frac12)\)  \(\approx\)  \(0.5735399128\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.30262701254731, −15.86716328445593, −15.41714141449524, −14.98686769432921, −14.18856244848622, −13.58265705266328, −13.08335972708972, −12.29146566437027, −12.07154469449418, −11.16092558507323, −10.84804777355415, −10.09416820242515, −9.480624551662315, −8.662493836080901, −8.282643264457376, −7.563976079680177, −6.934915352306940, −6.306634952300982, −5.601821071625448, −4.679339654217170, −4.106487319842271, −3.500591879925350, −2.566778784895122, −1.775522209513460, −0.3345148349639355, 0.3345148349639355, 1.775522209513460, 2.566778784895122, 3.500591879925350, 4.106487319842271, 4.679339654217170, 5.601821071625448, 6.306634952300982, 6.934915352306940, 7.563976079680177, 8.282643264457376, 8.662493836080901, 9.480624551662315, 10.09416820242515, 10.84804777355415, 11.16092558507323, 12.07154469449418, 12.29146566437027, 13.08335972708972, 13.58265705266328, 14.18856244848622, 14.98686769432921, 15.41714141449524, 15.86716328445593, 16.30262701254731

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