Properties

Label 4-33e2-1.1-c5e2-0-3
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $28.0123$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 18·3-s − 4-s + 58·5-s + 90·6-s − 286·7-s − 25·8-s + 243·9-s − 290·10-s − 242·11-s + 18·12-s − 166·13-s + 1.43e3·14-s − 1.04e3·15-s − 73·16-s − 800·17-s − 1.21e3·18-s − 1.47e3·19-s − 58·20-s + 5.14e3·21-s + 1.21e3·22-s − 3.37e3·23-s + 450·24-s + 698·25-s + 830·26-s − 2.91e3·27-s + 286·28-s + ⋯
L(s)  = 1  − 0.883·2-s − 1.15·3-s − 0.0312·4-s + 1.03·5-s + 1.02·6-s − 2.20·7-s − 0.138·8-s + 9-s − 0.917·10-s − 0.603·11-s + 0.0360·12-s − 0.272·13-s + 1.94·14-s − 1.19·15-s − 0.0712·16-s − 0.671·17-s − 0.883·18-s − 0.937·19-s − 0.0324·20-s + 2.54·21-s + 0.533·22-s − 1.32·23-s + 0.159·24-s + 0.223·25-s + 0.240·26-s − 0.769·27-s + 0.0689·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(28.0123\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1089,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{2} \)
11$C_1$ \( ( 1 + p^{2} T )^{2} \)
good2$C_2^2$ \( 1 + 5 T + 13 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 58 T + 2666 T^{2} - 58 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 286 T + 49638 T^{2} + 286 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 166 T + 685578 T^{2} + 166 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 800 T + 2840414 T^{2} + 800 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 1476 T + 5490470 T^{2} + 1476 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 3370 T + 9784358 T^{2} + 3370 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 6600 T + 44401126 T^{2} - 6600 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 7528 T + 66189630 T^{2} + 7528 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 29916 T + 361611230 T^{2} + 29916 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 5780 T + 217632230 T^{2} + 5780 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 16656 T + 264340262 T^{2} + 16656 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 7850 T + 191750726 T^{2} - 7850 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 14178 T + 671305114 T^{2} - 14178 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 17300 T + 1408269110 T^{2} - 17300 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 2946 T + 1451133506 T^{2} + 2946 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 31336 T + 2492616438 T^{2} - 31336 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 33810 T + 3469543750 T^{2} + 33810 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 60644 T + 2552552022 T^{2} - 60644 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 1870 T + 2176543686 T^{2} - 1870 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 58296 T + 101571026 p T^{2} + 58296 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 92388 T + 3271275766 T^{2} - 92388 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 7120 T - 2888428386 T^{2} - 7120 p^{5} T^{3} + p^{10} T^{4} \)
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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

−15.80915553911775117155067646633, −15.13360148874937302477256187810, −13.69607619656109181330460239955, −13.69412194241532301953161295221, −12.66574831874663081872849108495, −12.51797203314545027800527291420, −11.68869704591268006977706955129, −10.54104660329881261209451351712, −10.11923600090473649603053583263, −9.909161036763918665683027111396, −9.050934053527275779723401650026, −8.491670798461644446160255407481, −6.93799882683230437426918093224, −6.58098047892767702440617356094, −5.92849217179981441436906997408, −5.05448953737723395622846367728, −3.58137809057447407668855622201, −2.14021812507927974890337211927, 0, 0, 2.14021812507927974890337211927, 3.58137809057447407668855622201, 5.05448953737723395622846367728, 5.92849217179981441436906997408, 6.58098047892767702440617356094, 6.93799882683230437426918093224, 8.491670798461644446160255407481, 9.050934053527275779723401650026, 9.909161036763918665683027111396, 10.11923600090473649603053583263, 10.54104660329881261209451351712, 11.68869704591268006977706955129, 12.51797203314545027800527291420, 12.66574831874663081872849108495, 13.69412194241532301953161295221, 13.69607619656109181330460239955, 15.13360148874937302477256187810, 15.80915553911775117155067646633

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