# Properties

 Degree $4$ Conductor $81$ Sign $1$ Motivic weight $33$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.21e5·2-s + 6.13e9·4-s + 1.81e11·5-s − 6.71e13·7-s + 7.37e14·8-s + 2.20e16·10-s − 1.33e17·11-s − 2.98e18·13-s − 8.17e18·14-s + 8.99e19·16-s + 7.93e19·17-s − 1.36e21·19-s + 1.11e21·20-s − 1.62e22·22-s − 2.61e22·23-s − 1.95e23·25-s − 3.62e23·26-s − 4.12e23·28-s + 1.67e24·29-s − 6.23e24·31-s + 1.99e24·32-s + 9.65e24·34-s − 1.21e25·35-s − 1.04e26·37-s − 1.65e26·38-s + 1.33e26·40-s − 2.77e26·41-s + ⋯
 L(s)  = 1 + 1.31·2-s + 0.714·4-s + 0.530·5-s − 0.763·7-s + 0.926·8-s + 0.696·10-s − 0.878·11-s − 1.24·13-s − 1.00·14-s + 1.21·16-s + 0.395·17-s − 1.08·19-s + 0.379·20-s − 1.15·22-s − 0.889·23-s − 1.68·25-s − 1.63·26-s − 0.545·28-s + 1.24·29-s − 1.54·31-s + 0.290·32-s + 0.519·34-s − 0.405·35-s − 1.39·37-s − 1.42·38-s + 0.491·40-s − 0.679·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$81$$    =    $$3^{4}$$ Sign: $1$ Motivic weight: $$33$$ Character: induced by $\chi_{9} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 81,\ (\ :33/2, 33/2),\ 1)$$

## Particular Values

 $$L(17)$$ $$\approx$$ $$0.02121514055$$ $$L(\frac12)$$ $$\approx$$ $$0.02121514055$$ $$L(\frac{35}{2})$$ 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$D_{4}$ $$1 - 7605 p^{4} T + 2115925 p^{12} T^{2} - 7605 p^{37} T^{3} + p^{66} T^{4}$$
5$D_{4}$ $$1 - 1448492292 p^{3} T + 585507855025144558 p^{8} T^{2} - 1448492292 p^{36} T^{3} + p^{66} T^{4}$$
7$D_{4}$ $$1 + 9593297152400 p T +$$$$36\!\cdots\!50$$$$p^{4} T^{2} + 9593297152400 p^{34} T^{3} + p^{66} T^{4}$$
11$D_{4}$ $$1 + 12170165040174024 p T +$$$$23\!\cdots\!66$$$$p^{2} T^{2} + 12170165040174024 p^{34} T^{3} + p^{66} T^{4}$$
13$D_{4}$ $$1 + 229354652173803380 p T +$$$$61\!\cdots\!50$$$$p^{3} T^{2} + 229354652173803380 p^{34} T^{3} + p^{66} T^{4}$$
17$D_{4}$ $$1 - 79361149261175525340 T -$$$$26\!\cdots\!50$$$$p T^{2} - 79361149261175525340 p^{33} T^{3} + p^{66} T^{4}$$
19$D_{4}$ $$1 +$$$$13\!\cdots\!00$$$$T +$$$$17\!\cdots\!22$$$$p T^{2} +$$$$13\!\cdots\!00$$$$p^{33} T^{3} + p^{66} T^{4}$$
23$D_{4}$ $$1 +$$$$26\!\cdots\!40$$$$T +$$$$68\!\cdots\!50$$$$p T^{2} +$$$$26\!\cdots\!40$$$$p^{33} T^{3} + p^{66} T^{4}$$
29$D_{4}$ $$1 -$$$$57\!\cdots\!00$$$$p T +$$$$46\!\cdots\!58$$$$p^{2} T^{2} -$$$$57\!\cdots\!00$$$$p^{34} T^{3} + p^{66} T^{4}$$
31$D_{4}$ $$1 +$$$$20\!\cdots\!36$$$$p T +$$$$30\!\cdots\!86$$$$p^{2} T^{2} +$$$$20\!\cdots\!36$$$$p^{34} T^{3} + p^{66} T^{4}$$
37$D_{4}$ $$1 +$$$$10\!\cdots\!20$$$$T +$$$$13\!\cdots\!50$$$$T^{2} +$$$$10\!\cdots\!20$$$$p^{33} T^{3} + p^{66} T^{4}$$
41$D_{4}$ $$1 +$$$$27\!\cdots\!44$$$$T +$$$$22\!\cdots\!26$$$$T^{2} +$$$$27\!\cdots\!44$$$$p^{33} T^{3} + p^{66} T^{4}$$
43$D_{4}$ $$1 -$$$$15\!\cdots\!00$$$$T +$$$$12\!\cdots\!50$$$$T^{2} -$$$$15\!\cdots\!00$$$$p^{33} T^{3} + p^{66} T^{4}$$
47$D_{4}$ $$1 +$$$$11\!\cdots\!20$$$$p T +$$$$37\!\cdots\!50$$$$T^{2} +$$$$11\!\cdots\!20$$$$p^{34} T^{3} + p^{66} T^{4}$$
53$D_{4}$ $$1 -$$$$26\!\cdots\!20$$$$T +$$$$12\!\cdots\!50$$$$T^{2} -$$$$26\!\cdots\!20$$$$p^{33} T^{3} + p^{66} T^{4}$$
59$D_{4}$ $$1 -$$$$30\!\cdots\!00$$$$T +$$$$77\!\cdots\!58$$$$T^{2} -$$$$30\!\cdots\!00$$$$p^{33} T^{3} + p^{66} T^{4}$$
61$D_{4}$ $$1 +$$$$57\!\cdots\!36$$$$T +$$$$16\!\cdots\!86$$$$T^{2} +$$$$57\!\cdots\!36$$$$p^{33} T^{3} + p^{66} T^{4}$$
67$D_{4}$ $$1 -$$$$15\!\cdots\!60$$$$T +$$$$41\!\cdots\!50$$$$T^{2} -$$$$15\!\cdots\!60$$$$p^{33} T^{3} + p^{66} T^{4}$$
71$D_{4}$ $$1 -$$$$26\!\cdots\!76$$$$T +$$$$22\!\cdots\!66$$$$T^{2} -$$$$26\!\cdots\!76$$$$p^{33} T^{3} + p^{66} T^{4}$$
73$D_{4}$ $$1 -$$$$94\!\cdots\!40$$$$T +$$$$19\!\cdots\!50$$$$T^{2} -$$$$94\!\cdots\!40$$$$p^{33} T^{3} + p^{66} T^{4}$$
79$D_{4}$ $$1 +$$$$85\!\cdots\!00$$$$T +$$$$85\!\cdots\!78$$$$T^{2} +$$$$85\!\cdots\!00$$$$p^{33} T^{3} + p^{66} T^{4}$$
83$D_{4}$ $$1 +$$$$29\!\cdots\!20$$$$T +$$$$37\!\cdots\!50$$$$T^{2} +$$$$29\!\cdots\!20$$$$p^{33} T^{3} + p^{66} T^{4}$$
89$D_{4}$ $$1 +$$$$13\!\cdots\!00$$$$T +$$$$47\!\cdots\!38$$$$T^{2} +$$$$13\!\cdots\!00$$$$p^{33} T^{3} + p^{66} T^{4}$$
97$D_{4}$ $$1 +$$$$36\!\cdots\!60$$$$T +$$$$42\!\cdots\!50$$$$T^{2} +$$$$36\!\cdots\!60$$$$p^{33} T^{3} + p^{66} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$