↑

Properties

Label 2-11088-1.1-c1-0-30
Degree $2$
Conductor $11088$
Sign $-1$
Analytic cond. $88.5381$
Root an. cond. $9.40947$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 11-s − 4·13-s − 4·19-s + 4·23-s − 25-s − 2·29-s + 10·31-s − 2·35-s − 6·37-s + 4·43-s + 10·47-s + 49-s + 14·53-s − 2·55-s + 10·59-s − 8·61-s + 8·65-s − 8·67-s − 4·71-s + 4·73-s + 77-s − 16·79-s + 4·83-s − 10·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s − 0.338·35-s − 0.986·37-s + 0.609·43-s + 1.45·47-s + 1/7·49-s + 1.92·53-s − 0.269·55-s + 1.30·59-s − 1.02·61-s + 0.992·65-s − 0.977·67-s − 0.474·71-s + 0.468·73-s + 0.113·77-s − 1.80·79-s + 0.439·83-s − 1.05·89-s − 0.419·91-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

Degree: \(2\)
Conductor: \(11088\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(88.5381\)
Root analytic conductor: \(9.40947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11088,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.84152510693706, −16.18530301744859, −15.51247847849617, −14.99133846309854, −14.79237637202622, −13.85272512944902, −13.48939722744645, −12.48373667059060, −12.22581646145710, −11.63571092323463, −11.08847640840420, −10.35241320339812, −9.910983308024892, −8.907599222201657, −8.618756240625021, −7.799608880005398, −7.280605766385201, −6.765953323483856, −5.845679354550022, −5.121148707867814, −4.337252271337918, −3.990565243988865, −2.902562889336519, −2.252364407841245, −1.082554389871638, 0, 1.082554389871638, 2.252364407841245, 2.902562889336519, 3.990565243988865, 4.337252271337918, 5.121148707867814, 5.845679354550022, 6.765953323483856, 7.280605766385201, 7.799608880005398, 8.618756240625021, 8.907599222201657, 9.910983308024892, 10.35241320339812, 11.08847640840420, 11.63571092323463, 12.22581646145710, 12.48373667059060, 13.48939722744645, 13.85272512944902, 14.79237637202622, 14.99133846309854, 15.51247847849617, 16.18530301744859, 16.84152510693706

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