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 + 2·13-s − 6·17-s + 8·19-s + 4·23-s − 25-s − 2·29-s − 8·31-s − 2·35-s + 6·37-s − 6·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s − 2·55-s + 4·59-s − 14·61-s − 4·65-s + 4·67-s − 4·71-s − 14·73-s + 77-s + 8·79-s + 4·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 1.45·17-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 − 0.937·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s + 0.520·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s − 0.474·71-s − 1.63·73-s + 0.113·77-s + 0.900·79-s + 0.439·83-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.74073726960675, −15.96226300461547, −15.81194145102517, −15.02389446993866, −14.69754359574697, −13.83456003713443, −13.37099910849743, −12.85283430402046, −11.93157671460129, −11.57766896369497, −11.14342599509863, −10.57456541541796, −9.634642119626678, −9.052529265383199, −8.616951931820084, −7.643352133727088, −7.483031256436544, −6.633596769424717, −5.927254849785841, −5.036504508249632, −4.554759410967863, −3.629621983993562, −3.237177386965924, −2.048957245382733, −1.170604523763546, 0, 1.170604523763546, 2.048957245382733, 3.237177386965924, 3.629621983993562, 4.554759410967863, 5.036504508249632, 5.927254849785841, 6.633596769424717, 7.483031256436544, 7.643352133727088, 8.616951931820084, 9.052529265383199, 9.634642119626678, 10.57456541541796, 11.14342599509863, 11.57766896369497, 11.93157671460129, 12.85283430402046, 13.37099910849743, 13.83456003713443, 14.69754359574697, 15.02389446993866, 15.81194145102517, 15.96226300461547, 16.74073726960675

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