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  − 7-s − 11-s + 2·13-s − 2·19-s − 5·25-s + 6·29-s − 2·31-s + 2·37-s + 4·43-s − 6·47-s + 49-s + 6·53-s + 2·61-s + 4·67-s − 12·71-s − 4·73-s + 77-s − 8·79-s + 6·83-s + 6·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.301·11-s + 0.554·13-s − 0.458·19-s − 25-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.256·61-s + 0.488·67-s − 1.42·71-s − 0.468·73-s + 0.113·77-s − 0.900·79-s + 0.658·83-s + 0.635·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.67112562024177, −16.06806233587161, −15.83178528954987, −15.02805068682895, −14.59829125722420, −13.75889551342086, −13.43226596303616, −12.78498319876145, −12.22384504461522, −11.56337827403888, −11.00739817214610, −10.28978997665287, −9.893754132198377, −9.093247899182093, −8.544754272689191, −7.909313237492433, −7.246853438866235, −6.479567855019073, −5.978472986053235, −5.265578786624080, −4.406743808877434, −3.788682194639284, −2.960811092975216, −2.193502580258411, −1.171688489284412, 0, 1.171688489284412, 2.193502580258411, 2.960811092975216, 3.788682194639284, 4.406743808877434, 5.265578786624080, 5.978472986053235, 6.479567855019073, 7.246853438866235, 7.909313237492433, 8.544754272689191, 9.093247899182093, 9.893754132198377, 10.28978997665287, 11.00739817214610, 11.56337827403888, 12.22384504461522, 12.78498319876145, 13.43226596303616, 13.75889551342086, 14.59829125722420, 15.02805068682895, 15.83178528954987, 16.06806233587161, 16.67112562024177

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