# Properties

 Label 4-3e2-1.1-c27e2-0-0 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 $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2.15e4·2-s − 3.18e6·3-s + 1.99e8·4-s − 1.77e9·5-s − 6.88e10·6-s + 3.69e11·7-s + 1.47e12·8-s + 7.62e12·9-s − 3.82e13·10-s + 7.57e13·11-s − 6.37e14·12-s − 1.03e14·13-s + 7.97e15·14-s + 5.65e15·15-s + 2.32e16·16-s + 3.46e16·17-s + 1.64e17·18-s + 1.11e17·19-s − 3.54e17·20-s − 1.17e18·21-s + 1.63e18·22-s − 2.89e18·23-s − 4.70e18·24-s − 1.25e19·25-s − 2.22e18·26-s − 1.62e19·27-s + 7.39e19·28-s + ⋯
 L(s)  = 1 + 1.86·2-s − 1.15·3-s + 1.48·4-s − 0.649·5-s − 2.15·6-s + 1.44·7-s + 0.949·8-s + 9-s − 1.20·10-s + 0.661·11-s − 1.72·12-s − 0.0943·13-s + 2.68·14-s + 0.749·15-s + 1.28·16-s + 0.849·17-s + 1.86·18-s + 0.608·19-s − 0.967·20-s − 1.66·21-s + 1.23·22-s − 1.19·23-s − 1.09·24-s − 1.67·25-s − 0.175·26-s − 0.769·27-s + 2.14·28-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: $$0$$ Selberg data: $$(4,\ 9,\ (\ :27/2, 27/2),\ 1)$$

## Particular Values

 $$L(14)$$ $$\approx$$ $$5.048440310$$ $$L(\frac12)$$ $$\approx$$ $$5.048440310$$ $$L(\frac{29}{2})$$ 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$D_{4}$ $$1 - 10791 p T + 4153237 p^{6} T^{2} - 10791 p^{28} T^{3} + p^{54} T^{4}$$
5$D_{4}$ $$1 + 70877844 p^{2} T + 5007752027691686 p^{5} T^{2} + 70877844 p^{29} T^{3} + p^{54} T^{4}$$
7$D_{4}$ $$1 - 52809314272 p T +$$$$25\!\cdots\!94$$$$p^{2} T^{2} - 52809314272 p^{28} T^{3} + p^{54} T^{4}$$
11$D_{4}$ $$1 - 75762335668248 T +$$$$16\!\cdots\!14$$$$p^{2} T^{2} - 75762335668248 p^{27} T^{3} + p^{54} T^{4}$$
13$D_{4}$ $$1 + 7924698398276 p T +$$$$46\!\cdots\!54$$$$p^{2} T^{2} + 7924698398276 p^{28} T^{3} + p^{54} T^{4}$$
17$D_{4}$ $$1 - 2040683034067524 p T +$$$$29\!\cdots\!58$$$$p^{2} T^{2} - 2040683034067524 p^{28} T^{3} + p^{54} T^{4}$$
19$D_{4}$ $$1 - 5874761924139064 p T +$$$$36\!\cdots\!82$$$$p T^{2} - 5874761924139064 p^{28} T^{3} + p^{54} T^{4}$$
23$D_{4}$ $$1 + 125771438079103824 p T +$$$$18\!\cdots\!26$$$$p^{2} T^{2} + 125771438079103824 p^{28} T^{3} + p^{54} T^{4}$$
29$D_{4}$ $$1 - 29959552473322806972 T +$$$$45\!\cdots\!78$$$$T^{2} - 29959552473322806972 p^{27} T^{3} + p^{54} T^{4}$$
31$D_{4}$ $$1 - 10367463257055494032 T +$$$$26\!\cdots\!22$$$$T^{2} - 10367463257055494032 p^{27} T^{3} + p^{54} T^{4}$$
37$D_{4}$ $$1 +$$$$37\!\cdots\!16$$$$T +$$$$78\!\cdots\!86$$$$T^{2} +$$$$37\!\cdots\!16$$$$p^{27} T^{3} + p^{54} T^{4}$$
41$D_{4}$ $$1 -$$$$14\!\cdots\!72$$$$T +$$$$35\!\cdots\!42$$$$T^{2} -$$$$14\!\cdots\!72$$$$p^{27} T^{3} + p^{54} T^{4}$$
43$D_{4}$ $$1 -$$$$97\!\cdots\!40$$$$T +$$$$26\!\cdots\!90$$$$T^{2} -$$$$97\!\cdots\!40$$$$p^{27} T^{3} + p^{54} T^{4}$$
47$D_{4}$ $$1 -$$$$89\!\cdots\!20$$$$T +$$$$44\!\cdots\!90$$$$T^{2} -$$$$89\!\cdots\!20$$$$p^{27} T^{3} + p^{54} T^{4}$$
53$D_{4}$ $$1 -$$$$42\!\cdots\!48$$$$T +$$$$11\!\cdots\!94$$$$T^{2} -$$$$42\!\cdots\!48$$$$p^{27} T^{3} + p^{54} T^{4}$$
59$D_{4}$ $$1 -$$$$36\!\cdots\!24$$$$T +$$$$55\!\cdots\!38$$$$T^{2} -$$$$36\!\cdots\!24$$$$p^{27} T^{3} + p^{54} T^{4}$$
61$D_{4}$ $$1 -$$$$92\!\cdots\!20$$$$T +$$$$32\!\cdots\!58$$$$T^{2} -$$$$92\!\cdots\!20$$$$p^{27} T^{3} + p^{54} T^{4}$$
67$D_{4}$ $$1 -$$$$41\!\cdots\!68$$$$T +$$$$44\!\cdots\!02$$$$T^{2} -$$$$41\!\cdots\!68$$$$p^{27} T^{3} + p^{54} T^{4}$$
71$D_{4}$ $$1 -$$$$97\!\cdots\!84$$$$T +$$$$15\!\cdots\!46$$$$T^{2} -$$$$97\!\cdots\!84$$$$p^{27} T^{3} + p^{54} T^{4}$$
73$D_{4}$ $$1 -$$$$28\!\cdots\!88$$$$T +$$$$56\!\cdots\!14$$$$T^{2} -$$$$28\!\cdots\!88$$$$p^{27} T^{3} + p^{54} T^{4}$$
79$D_{4}$ $$1 +$$$$49\!\cdots\!20$$$$T +$$$$34\!\cdots\!18$$$$T^{2} +$$$$49\!\cdots\!20$$$$p^{27} T^{3} + p^{54} T^{4}$$
83$D_{4}$ $$1 +$$$$12\!\cdots\!36$$$$T +$$$$16\!\cdots\!82$$$$T^{2} +$$$$12\!\cdots\!36$$$$p^{27} T^{3} + p^{54} T^{4}$$
89$D_{4}$ $$1 -$$$$20\!\cdots\!56$$$$T +$$$$40\!\cdots\!38$$$$T^{2} -$$$$20\!\cdots\!56$$$$p^{27} T^{3} + p^{54} T^{4}$$
97$D_{4}$ $$1 +$$$$14\!\cdots\!72$$$$T +$$$$13\!\cdots\!22$$$$T^{2} +$$$$14\!\cdots\!72$$$$p^{27} T^{3} + p^{54} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$