Properties

Label 4-33e2-1.1-c1e2-0-1
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $0.0694355$
Root an. cond. $0.513328$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·4-s − 2·9-s − 4·12-s + 12·16-s − 25-s − 5·27-s + 10·31-s + 8·36-s − 14·37-s + 12·48-s + 14·49-s − 32·64-s − 26·67-s − 75-s + 81-s + 10·93-s + 34·97-s + 4·100-s − 8·103-s + 20·108-s − 14·111-s − 11·121-s − 40·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s − 2/3·9-s − 1.15·12-s + 3·16-s − 1/5·25-s − 0.962·27-s + 1.79·31-s + 4/3·36-s − 2.30·37-s + 1.73·48-s + 2·49-s − 4·64-s − 3.17·67-s − 0.115·75-s + 1/9·81-s + 1.03·93-s + 3.45·97-s + 2/5·100-s − 0.788·103-s + 1.92·108-s − 1.32·111-s − 121-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.0694355\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1089,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4460885105\)
\(L(\frac12)\) \(\approx\) \(0.4460885105\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.17113961041653810289302193238, −16.85574838471113963425433979456, −15.74352154365985779192189445348, −15.15355040523506862802895652520, −14.48010355709446529915257600066, −14.04167765341748268011552322659, −13.52133580755889374274588661744, −13.25737799010645723889294192930, −12.18377057708802742812991424989, −11.91285142562008747901103505623, −10.50315208485772406638675869216, −10.14183943032400540896029256073, −9.160132383755029723980325860329, −8.841889407311114617556569512502, −8.264947596125555144370168258865, −7.48410396012403277235033293890, −6.01987084005356094215710711834, −5.18130267836277930636751198273, −4.27630887656717266868263341494, −3.25699087747366984275342802206, 3.25699087747366984275342802206, 4.27630887656717266868263341494, 5.18130267836277930636751198273, 6.01987084005356094215710711834, 7.48410396012403277235033293890, 8.264947596125555144370168258865, 8.841889407311114617556569512502, 9.160132383755029723980325860329, 10.14183943032400540896029256073, 10.50315208485772406638675869216, 11.91285142562008747901103505623, 12.18377057708802742812991424989, 13.25737799010645723889294192930, 13.52133580755889374274588661744, 14.04167765341748268011552322659, 14.48010355709446529915257600066, 15.15355040523506862802895652520, 15.74352154365985779192189445348, 16.85574838471113963425433979456, 17.17113961041653810289302193238

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