# Properties

 Label 4-3e2-1.1-c27e2-0-1 Degree $4$ Conductor $9$ Sign $1$ Analytic cond. $191.979$ Root an. cond. $3.72232$ Motivic weight $27$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3.16e3·2-s + 3.18e6·3-s − 1.26e8·4-s − 4.90e9·5-s + 1.01e10·6-s − 1.51e11·7-s − 4.09e11·8-s + 7.62e12·9-s − 1.55e13·10-s + 4.08e13·11-s − 4.04e14·12-s − 4.17e14·13-s − 4.80e14·14-s − 1.56e16·15-s − 6.23e14·16-s − 7.37e16·17-s + 2.41e16·18-s − 3.46e17·19-s + 6.21e17·20-s − 4.83e17·21-s + 1.29e17·22-s + 2.49e18·23-s − 1.30e18·24-s + 8.31e18·25-s − 1.32e18·26-s + 1.62e19·27-s + 1.92e19·28-s + ⋯
 L(s)  = 1 + 0.273·2-s + 1.15·3-s − 0.944·4-s − 1.79·5-s + 0.315·6-s − 0.591·7-s − 0.263·8-s + 9-s − 0.491·10-s + 0.356·11-s − 1.09·12-s − 0.382·13-s − 0.161·14-s − 2.07·15-s − 0.0346·16-s − 1.80·17-s + 0.273·18-s − 1.89·19-s + 1.69·20-s − 0.683·21-s + 0.0975·22-s + 1.03·23-s − 0.304·24-s + 1.11·25-s − 0.104·26-s + 0.769·27-s + 0.558·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $1$ Analytic conductor: $$191.979$$ Root analytic conductor: $$3.72232$$ Motivic weight: $$27$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 9,\ (\ :27/2, 27/2),\ 1)$$

## Particular Values

 $$L(14)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{29}{2})$$ 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$C_1$ $$( 1 - p^{13} T )^{2}$$
good2$D_{4}$ $$1 - 99 p^{5} T + 133597 p^{10} T^{2} - 99 p^{32} T^{3} + p^{54} T^{4}$$
5$D_{4}$ $$1 + 981213012 p T + 25204374933470302 p^{4} T^{2} + 981213012 p^{28} T^{3} + p^{54} T^{4}$$
7$D_{4}$ $$1 + 21665298512 p T +$$$$13\!\cdots\!34$$$$p^{2} T^{2} + 21665298512 p^{28} T^{3} + p^{54} T^{4}$$
11$D_{4}$ $$1 - 3714090193368 p T +$$$$97\!\cdots\!54$$$$p^{2} T^{2} - 3714090193368 p^{28} T^{3} + p^{54} T^{4}$$
13$D_{4}$ $$1 + 32107530664484 p T +$$$$11\!\cdots\!34$$$$p^{2} T^{2} + 32107530664484 p^{28} T^{3} + p^{54} T^{4}$$
17$D_{4}$ $$1 + 73795652103164508 T +$$$$27\!\cdots\!86$$$$p T^{2} + 73795652103164508 p^{27} T^{3} + p^{54} T^{4}$$
19$D_{4}$ $$1 + 346903482355728584 T +$$$$40\!\cdots\!82$$$$p T^{2} + 346903482355728584 p^{27} T^{3} + p^{54} T^{4}$$
23$D_{4}$ $$1 - 2497625277930684432 T +$$$$48\!\cdots\!58$$$$p T^{2} - 2497625277930684432 p^{27} T^{3} + p^{54} T^{4}$$
29$D_{4}$ $$1 + 66866817822024885588 T +$$$$64\!\cdots\!38$$$$T^{2} + 66866817822024885588 p^{27} T^{3} + p^{54} T^{4}$$
31$D_{4}$ $$1 + 4237787486695403168 p T +$$$$32\!\cdots\!22$$$$T^{2} + 4237787486695403168 p^{28} T^{3} + p^{54} T^{4}$$
37$D_{4}$ $$1 +$$$$24\!\cdots\!04$$$$T +$$$$19\!\cdots\!86$$$$T^{2} +$$$$24\!\cdots\!04$$$$p^{27} T^{3} + p^{54} T^{4}$$
41$D_{4}$ $$1 -$$$$50\!\cdots\!12$$$$T +$$$$66\!\cdots\!22$$$$T^{2} -$$$$50\!\cdots\!12$$$$p^{27} T^{3} + p^{54} T^{4}$$
43$D_{4}$ $$1 +$$$$35\!\cdots\!20$$$$T +$$$$57\!\cdots\!90$$$$T^{2} +$$$$35\!\cdots\!20$$$$p^{27} T^{3} + p^{54} T^{4}$$
47$D_{4}$ $$1 -$$$$48\!\cdots\!80$$$$T +$$$$15\!\cdots\!90$$$$T^{2} -$$$$48\!\cdots\!80$$$$p^{27} T^{3} + p^{54} T^{4}$$
53$D_{4}$ $$1 -$$$$35\!\cdots\!92$$$$T +$$$$10\!\cdots\!74$$$$T^{2} -$$$$35\!\cdots\!92$$$$p^{27} T^{3} + p^{54} T^{4}$$
59$D_{4}$ $$1 +$$$$15\!\cdots\!76$$$$T +$$$$17\!\cdots\!58$$$$T^{2} +$$$$15\!\cdots\!76$$$$p^{27} T^{3} + p^{54} T^{4}$$
61$D_{4}$ $$1 +$$$$29\!\cdots\!00$$$$T +$$$$30\!\cdots\!98$$$$T^{2} +$$$$29\!\cdots\!00$$$$p^{27} T^{3} + p^{54} T^{4}$$
67$D_{4}$ $$1 +$$$$14\!\cdots\!88$$$$T +$$$$40\!\cdots\!82$$$$T^{2} +$$$$14\!\cdots\!88$$$$p^{27} T^{3} + p^{54} T^{4}$$
71$D_{4}$ $$1 +$$$$18\!\cdots\!76$$$$T +$$$$23\!\cdots\!26$$$$T^{2} +$$$$18\!\cdots\!76$$$$p^{27} T^{3} + p^{54} T^{4}$$
73$D_{4}$ $$1 +$$$$18\!\cdots\!88$$$$T +$$$$33\!\cdots\!54$$$$T^{2} +$$$$18\!\cdots\!88$$$$p^{27} T^{3} + p^{54} T^{4}$$
79$D_{4}$ $$1 +$$$$29\!\cdots\!40$$$$T +$$$$21\!\cdots\!18$$$$T^{2} +$$$$29\!\cdots\!40$$$$p^{27} T^{3} + p^{54} T^{4}$$
83$D_{4}$ $$1 -$$$$68\!\cdots\!36$$$$T +$$$$13\!\cdots\!22$$$$T^{2} -$$$$68\!\cdots\!36$$$$p^{27} T^{3} + p^{54} T^{4}$$
89$D_{4}$ $$1 -$$$$52\!\cdots\!36$$$$T +$$$$14\!\cdots\!98$$$$T^{2} -$$$$52\!\cdots\!36$$$$p^{27} T^{3} + p^{54} T^{4}$$
97$D_{4}$ $$1 -$$$$79\!\cdots\!12$$$$T +$$$$68\!\cdots\!62$$$$T^{2} -$$$$79\!\cdots\!12$$$$p^{27} T^{3} + p^{54} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$