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  − 3·5-s − 7-s + 11-s + 5·13-s − 6·17-s + 19-s + 4·25-s + 9·29-s − 8·31-s + 3·35-s − 7·37-s − 6·41-s + 10·43-s + 3·47-s + 49-s − 6·53-s − 3·55-s − 9·59-s + 2·61-s − 15·65-s + 13·67-s + 6·71-s + 11·73-s − 77-s + 10·79-s − 6·83-s + 18·85-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 0.301·11-s + 1.38·13-s − 1.45·17-s + 0.229·19-s + 4/5·25-s + 1.67·29-s − 1.43·31-s + 0.507·35-s − 1.15·37-s − 0.937·41-s + 1.52·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.404·55-s − 1.17·59-s + 0.256·61-s − 1.86·65-s + 1.58·67-s + 0.712·71-s + 1.28·73-s − 0.113·77-s + 1.12·79-s − 0.658·83-s + 1.95·85-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.64106611051500, −15.92396067221983, −15.65041507614632, −15.44933501125458, −14.49726677830986, −13.84295626943906, −13.46017120469332, −12.46842411252563, −12.37432300786523, −11.48078064803765, −10.93487390924330, −10.75779564211861, −9.666097725875053, −8.996520447686158, −8.487519304016985, −8.021871279159479, −7.115614086092751, −6.700900355830849, −6.038241890567920, −5.089015272349852, −4.313185272580107, −3.749446976551089, −3.240014334261831, −2.151231182729000, −1.022575916296861, 0, 1.022575916296861, 2.151231182729000, 3.240014334261831, 3.749446976551089, 4.313185272580107, 5.089015272349852, 6.038241890567920, 6.700900355830849, 7.115614086092751, 8.021871279159479, 8.487519304016985, 8.996520447686158, 9.666097725875053, 10.75779564211861, 10.93487390924330, 11.48078064803765, 12.37432300786523, 12.46842411252563, 13.46017120469332, 13.84295626943906, 14.49726677830986, 15.44933501125458, 15.65041507614632, 15.92396067221983, 16.64106611051500

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