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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 11-s + 3·13-s − 2·17-s + 5·19-s − 8·23-s − 4·25-s − 7·29-s + 8·31-s + 35-s − 3·37-s + 10·41-s + 10·43-s + 7·47-s + 49-s + 2·53-s + 55-s − 9·59-s − 2·61-s − 3·65-s + 3·67-s + 6·71-s + 73-s + 77-s − 10·79-s − 6·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s + 0.832·13-s − 0.485·17-s + 1.14·19-s − 1.66·23-s − 4/5·25-s − 1.29·29-s + 1.43·31-s + 0.169·35-s − 0.493·37-s + 1.56·41-s + 1.52·43-s + 1.02·47-s + 1/7·49-s + 0.274·53-s + 0.134·55-s − 1.17·59-s − 0.256·61-s − 0.372·65-s + 0.366·67-s + 0.712·71-s + 0.117·73-s + 0.113·77-s − 1.12·79-s − 0.658·83-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 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.65958818885075, −15.95065283409461, −15.73559840651362, −15.37846219146380, −14.34835657179876, −13.84963474166167, −13.51303822391615, −12.67017274419474, −12.20915083216691, −11.55032045123183, −11.07482694342111, −10.38932095223075, −9.729733825422263, −9.201490075393919, −8.501314566317073, −7.642905162228815, −7.565925220277237, −6.443515504274846, −5.920811175847233, −5.359665035204849, −4.124876311079234, −4.006815159492995, −2.982697601501780, −2.208306052413562, −1.110452087756801, 0, 1.110452087756801, 2.208306052413562, 2.982697601501780, 4.006815159492995, 4.124876311079234, 5.359665035204849, 5.920811175847233, 6.443515504274846, 7.565925220277237, 7.642905162228815, 8.501314566317073, 9.201490075393919, 9.729733825422263, 10.38932095223075, 11.07482694342111, 11.55032045123183, 12.20915083216691, 12.67017274419474, 13.51303822391615, 13.84963474166167, 14.34835657179876, 15.37846219146380, 15.73559840651362, 15.95065283409461, 16.65958818885075

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