# Properties

 Label 4-33e2-1.1-c5e2-0-0 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 $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 18·3-s + 15·4-s − 38·5-s − 18·6-s − 18·7-s + 61·8-s + 243·9-s − 38·10-s + 242·11-s − 270·12-s − 66·13-s − 18·14-s + 684·15-s − 721·16-s − 920·17-s + 243·18-s − 2.93e3·19-s − 570·20-s + 324·21-s + 242·22-s + 5.24e3·23-s − 1.09e3·24-s + 2.65e3·25-s − 66·26-s − 2.91e3·27-s − 270·28-s + ⋯
 L(s)  = 1 + 0.176·2-s − 1.15·3-s + 0.468·4-s − 0.679·5-s − 0.204·6-s − 0.138·7-s + 0.336·8-s + 9-s − 0.120·10-s + 0.603·11-s − 0.541·12-s − 0.108·13-s − 0.0245·14-s + 0.784·15-s − 0.704·16-s − 0.772·17-s + 0.176·18-s − 1.86·19-s − 0.318·20-s + 0.160·21-s + 0.106·22-s + 2.06·23-s − 0.389·24-s + 0.850·25-s − 0.0191·26-s − 0.769·27-s − 0.0650·28-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: $$0$$ Selberg data: $$(4,\ 1089,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.194348967$$ $$L(\frac12)$$ $$\approx$$ $$1.194348967$$ $$L(\frac{7}{2})$$ 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$D_{4}$ $$1 - T - 7 p T^{2} - p^{5} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 + 38 T - 1214 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 + 18 T + 33382 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 66 T - 135542 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 920 T + 3050062 T^{2} + 920 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 2932 T + 7100102 T^{2} + 2932 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 5246 T + 19699918 T^{2} - 5246 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 12600 T + 79348870 T^{2} + 12600 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 9936 T + 53013118 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 5996 T + 139663118 T^{2} - 5996 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 24244 T + 337184038 T^{2} - 24244 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 - 20360 T + 277332086 T^{2} - 20360 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 5806 T + 419753950 T^{2} + 5806 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 40770 T + 1224329794 T^{2} - 40770 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 18212 T + 455377462 T^{2} - 18212 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 11398 T + 1131625826 T^{2} + 11398 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 65368 T + 3722340342 T^{2} - 65368 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 61446 T + 3799408318 T^{2} - 61446 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 53412 T + 4851340822 T^{2} - 53412 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 17122 T + 5872391094 T^{2} - 17122 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 14304 T + 6955271542 T^{2} + 14304 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 58140 T + 11297620726 T^{2} + 58140 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 183056 T + 25458744990 T^{2} + 183056 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$