Properties

Label 4-33e2-1.1-c3e2-0-2
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $3.79105$
Root an. cond. $1.39537$
Motivic weight $3$
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  + 2-s + 6·3-s − 7·4-s + 16·5-s + 6·6-s + 2·7-s − 7·8-s + 27·9-s + 16·10-s + 22·11-s − 42·12-s − 76·13-s + 2·14-s + 96·15-s − 7·16-s − 26·17-s + 27·18-s − 54·19-s − 112·20-s + 12·21-s + 22·22-s + 224·23-s − 42·24-s + 74·25-s − 76·26-s + 108·27-s − 14·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.15·3-s − 7/8·4-s + 1.43·5-s + 0.408·6-s + 0.107·7-s − 0.309·8-s + 9-s + 0.505·10-s + 0.603·11-s − 1.01·12-s − 1.62·13-s + 0.0381·14-s + 1.65·15-s − 0.109·16-s − 0.370·17-s + 0.353·18-s − 0.652·19-s − 1.25·20-s + 0.124·21-s + 0.213·22-s + 2.03·23-s − 0.357·24-s + 0.591·25-s − 0.573·26-s + 0.769·27-s − 0.0944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.79105\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1089,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.365410141\)
\(L(\frac12)\) \(\approx\) \(2.365410141\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
11$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 - T + p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 16 T + 182 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 654 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 76 T + 5310 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 54 T + 11774 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 112 T + p^{3} T^{2} )^{2} \)
29$D_{4}$ \( 1 - 222 T + 43642 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 48 T + 85910 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 66 T + 99086 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 64 T + 189662 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 84 T + 164350 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 196 T + p^{3} T^{2} )^{2} \)
61$D_{4}$ \( 1 + 1104 T + 736358 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 928 T + 626214 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 456 T + 488494 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 592 T + 341742 T^{2} + 592 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 230 T + 954126 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 348 T + 307798 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \)
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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

−16.77078261848685652141940426447, −15.72197703177180852690966096402, −14.80222814541924372359390981129, −14.79029922420424043623765682971, −13.87430331569316197081791324882, −13.83824080161561347647576265176, −12.94652457996615243548870646512, −12.78081106325742402964963034825, −11.76900512853269557218188131478, −10.67719953626078168455926542217, −9.866108343323399937237525599678, −9.522932689938979383729287264358, −8.904018914491728706386586193173, −8.327450473960769027452790593869, −7.10836220703054412540017612900, −6.49310939166068878234154025260, −5.03136440772887571564301745924, −4.66921356206518403831190459786, −3.15974324457558183552847942675, −1.97220520084630365736708225749, 1.97220520084630365736708225749, 3.15974324457558183552847942675, 4.66921356206518403831190459786, 5.03136440772887571564301745924, 6.49310939166068878234154025260, 7.10836220703054412540017612900, 8.327450473960769027452790593869, 8.904018914491728706386586193173, 9.522932689938979383729287264358, 9.866108343323399937237525599678, 10.67719953626078168455926542217, 11.76900512853269557218188131478, 12.78081106325742402964963034825, 12.94652457996615243548870646512, 13.83824080161561347647576265176, 13.87430331569316197081791324882, 14.79029922420424043623765682971, 14.80222814541924372359390981129, 15.72197703177180852690966096402, 16.77078261848685652141940426447

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