Properties

Label 2-11088-1.1-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 11-s + 6·13-s − 6·17-s − 8·23-s − 25-s + 2·29-s − 4·31-s − 2·35-s − 2·37-s + 2·41-s + 12·43-s + 12·47-s + 49-s − 6·53-s − 2·55-s + 4·59-s + 6·61-s − 12·65-s − 4·67-s − 2·73-s + 77-s − 8·79-s + 12·85-s + 6·89-s + 6·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.301·11-s + 1.66·13-s − 1.45·17-s − 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s + 0.768·61-s − 1.48·65-s − 0.488·67-s − 0.234·73-s + 0.113·77-s − 0.900·79-s + 1.30·85-s + 0.635·89-s + 0.628·91-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: \(0\)
Selberg data: \((2,\ 11088,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579031123\)
\(L(\frac12)\) \(\approx\) \(1.579031123\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.11502086550629, −15.86118291421270, −15.66759451252851, −14.80605348949853, −14.14709744198393, −13.71968385217766, −13.08140369384830, −12.39982310840207, −11.77901194824096, −11.29339528696083, −10.82765739405334, −10.26629163859589, −9.204624442018878, −8.822295188148051, −8.206256907762717, −7.648696668615248, −6.963224711565187, −6.123531471818431, −5.770411328766132, −4.603674733702279, −3.990874585829314, −3.722352793025873, −2.496407378647474, −1.679598128453642, −0.5879239591642018, 0.5879239591642018, 1.679598128453642, 2.496407378647474, 3.722352793025873, 3.990874585829314, 4.603674733702279, 5.770411328766132, 6.123531471818431, 6.963224711565187, 7.648696668615248, 8.206256907762717, 8.822295188148051, 9.204624442018878, 10.26629163859589, 10.82765739405334, 11.29339528696083, 11.77901194824096, 12.39982310840207, 13.08140369384830, 13.71968385217766, 14.14709744198393, 14.80605348949853, 15.66759451252851, 15.86118291421270, 16.11502086550629

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