# Properties

 Label 4-3e4-1.1-c21e2-0-0 Degree $4$ Conductor $81$ Sign $1$ Analytic cond. $632.671$ Root an. cond. $5.01527$ Motivic weight $21$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 666·2-s − 1.28e6·4-s − 9.96e5·5-s + 6.79e8·7-s + 6.11e8·8-s + 6.63e8·10-s − 2.19e11·11-s − 4.84e10·13-s − 4.52e11·14-s − 1.65e12·16-s + 1.13e13·17-s + 1.19e13·19-s + 1.28e12·20-s + 1.46e14·22-s + 1.46e14·23-s − 4.78e14·25-s + 3.22e13·26-s − 8.74e14·28-s + 1.79e15·29-s + 1.11e16·31-s + 4.76e15·32-s − 7.54e15·34-s − 6.77e14·35-s + 1.27e16·37-s − 7.96e15·38-s − 6.09e14·40-s − 1.22e17·41-s + ⋯
 L(s)  = 1 − 0.459·2-s − 0.613·4-s − 0.0456·5-s + 0.909·7-s + 0.201·8-s + 0.0209·10-s − 2.55·11-s − 0.0975·13-s − 0.418·14-s − 0.375·16-s + 1.36·17-s + 0.447·19-s + 0.0279·20-s + 1.17·22-s + 0.737·23-s − 1.00·25-s + 0.0448·26-s − 0.557·28-s + 0.793·29-s + 2.44·31-s + 0.748·32-s − 0.627·34-s − 0.0415·35-s + 0.435·37-s − 0.205·38-s − 0.00918·40-s − 1.43·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$81$$    =    $$3^{4}$$ Sign: $1$ Analytic conductor: $$632.671$$ Root analytic conductor: $$5.01527$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 81,\ (\ :21/2, 21/2),\ 1)$$

## Particular Values

 $$L(11)$$ $$\approx$$ $$0.9685226229$$ $$L(\frac12)$$ $$\approx$$ $$0.9685226229$$ $$L(\frac{23}{2})$$ 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 $$1$$
good2$D_{4}$ $$1 + 333 p T + 54041 p^{5} T^{2} + 333 p^{22} T^{3} + p^{42} T^{4}$$
5$D_{4}$ $$1 + 996876 T + 19189088648254 p^{2} T^{2} + 996876 p^{21} T^{3} + p^{42} T^{4}$$
7$D_{4}$ $$1 - 97128016 p T + 23414651512456686 p^{2} T^{2} - 97128016 p^{22} T^{3} + p^{42} T^{4}$$
11$D_{4}$ $$1 + 19988102088 p T +$$$$20\!\cdots\!54$$$$p^{2} T^{2} + 19988102088 p^{22} T^{3} + p^{42} T^{4}$$
13$D_{4}$ $$1 + 48468909956 T -$$$$66\!\cdots\!82$$$$p T^{2} + 48468909956 p^{21} T^{3} + p^{42} T^{4}$$
17$D_{4}$ $$1 - 666678178908 p T +$$$$56\!\cdots\!22$$$$p^{2} T^{2} - 666678178908 p^{22} T^{3} + p^{42} T^{4}$$
19$D_{4}$ $$1 - 629504474296 p T +$$$$48\!\cdots\!38$$$$p^{2} T^{2} - 629504474296 p^{22} T^{3} + p^{42} T^{4}$$
23$D_{4}$ $$1 - 146508390063504 T +$$$$70\!\cdots\!46$$$$T^{2} - 146508390063504 p^{21} T^{3} + p^{42} T^{4}$$
29$D_{4}$ $$1 - 1798520043674052 T +$$$$75\!\cdots\!58$$$$T^{2} - 1798520043674052 p^{21} T^{3} + p^{42} T^{4}$$
31$D_{4}$ $$1 - 11169107526944992 T +$$$$65\!\cdots\!62$$$$T^{2} - 11169107526944992 p^{21} T^{3} + p^{42} T^{4}$$
37$D_{4}$ $$1 - 12736264858660012 T +$$$$11\!\cdots\!54$$$$T^{2} - 12736264858660012 p^{21} T^{3} + p^{42} T^{4}$$
41$D_{4}$ $$1 + 122972020616468052 T +$$$$11\!\cdots\!02$$$$T^{2} + 122972020616468052 p^{21} T^{3} + p^{42} T^{4}$$
43$D_{4}$ $$1 - 288455418162270040 T +$$$$60\!\cdots\!30$$$$T^{2} - 288455418162270040 p^{21} T^{3} + p^{42} T^{4}$$
47$D_{4}$ $$1 + 837243745741596960 T +$$$$43\!\cdots\!10$$$$T^{2} + 837243745741596960 p^{21} T^{3} + p^{42} T^{4}$$
53$D_{4}$ $$1 - 43007964012775764 T +$$$$30\!\cdots\!46$$$$T^{2} - 43007964012775764 p^{21} T^{3} + p^{42} T^{4}$$
59$D_{4}$ $$1 - 3523823330903857224 T +$$$$22\!\cdots\!98$$$$T^{2} - 3523823330903857224 p^{21} T^{3} + p^{42} T^{4}$$
61$D_{4}$ $$1 + 1779023128451013860 T +$$$$54\!\cdots\!38$$$$T^{2} + 1779023128451013860 p^{21} T^{3} + p^{42} T^{4}$$
67$D_{4}$ $$1 + 16454068667621610296 T +$$$$48\!\cdots\!38$$$$T^{2} + 16454068667621610296 p^{21} T^{3} + p^{42} T^{4}$$
71$D_{4}$ $$1 + 17379227131150420944 T +$$$$15\!\cdots\!26$$$$T^{2} + 17379227131150420944 p^{21} T^{3} + p^{42} T^{4}$$
73$D_{4}$ $$1 - 50891146268473989076 T +$$$$15\!\cdots\!06$$$$T^{2} - 50891146268473989076 p^{21} T^{3} + p^{42} T^{4}$$
79$D_{4}$ $$1 + 54055785594190591040 T +$$$$14\!\cdots\!58$$$$T^{2} + 54055785594190591040 p^{21} T^{3} + p^{42} T^{4}$$
83$D_{4}$ $$1 +$$$$11\!\cdots\!88$$$$T +$$$$39\!\cdots\!78$$$$T^{2} +$$$$11\!\cdots\!88$$$$p^{21} T^{3} + p^{42} T^{4}$$
89$D_{4}$ $$1 +$$$$22\!\cdots\!24$$$$T +$$$$17\!\cdots\!98$$$$T^{2} +$$$$22\!\cdots\!24$$$$p^{21} T^{3} + p^{42} T^{4}$$
97$D_{4}$ $$1 +$$$$12\!\cdots\!36$$$$T +$$$$78\!\cdots\!18$$$$T^{2} +$$$$12\!\cdots\!36$$$$p^{21} T^{3} + p^{42} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$