Properties

Label 2-11088-1.1-c1-0-23
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  − 3·5-s − 7-s − 11-s − 13-s − 5·19-s + 6·23-s + 4·25-s + 3·29-s + 4·31-s + 3·35-s − 7·37-s + 12·41-s − 2·43-s + 3·47-s + 49-s − 6·53-s + 3·55-s + 3·59-s + 2·61-s + 3·65-s + 67-s + 12·71-s − 7·73-s + 77-s + 4·79-s − 6·89-s + 91-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s − 1.14·19-s + 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 1.87·41-s − 0.304·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.404·55-s + 0.390·59-s + 0.256·61-s + 0.372·65-s + 0.122·67-s + 1.42·71-s − 0.819·73-s + 0.113·77-s + 0.450·79-s − 0.635·89-s + 0.104·91-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−16.74390180178843, −16.02028725075063, −15.67451194824048, −15.18271683546963, −14.62958899106597, −13.99851833634735, −13.19900408188817, −12.65192254492612, −12.26043720017795, −11.59667300865559, −10.94824175029229, −10.58313682041843, −9.767976933004637, −9.044040974557456, −8.428000185066308, −7.938064608061662, −7.209157287584483, −6.743452908745466, −5.964450815600656, −5.021787680126356, −4.450916464333543, −3.780911727544894, −3.047038071966626, −2.305971383356267, −0.9459250541200533, 0, 0.9459250541200533, 2.305971383356267, 3.047038071966626, 3.780911727544894, 4.450916464333543, 5.021787680126356, 5.964450815600656, 6.743452908745466, 7.209157287584483, 7.938064608061662, 8.428000185066308, 9.044040974557456, 9.767976933004637, 10.58313682041843, 10.94824175029229, 11.59667300865559, 12.26043720017795, 12.65192254492612, 13.19900408188817, 13.99851833634735, 14.62958899106597, 15.18271683546963, 15.67451194824048, 16.02028725075063, 16.74390180178843

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