# Properties

 Degree $4$ Conductor $9$ Sign $1$ Motivic weight $35$ Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6.09e4·2-s + 2.58e8·3-s − 3.93e9·4-s − 1.33e12·5-s − 1.57e13·6-s − 1.20e15·7-s − 1.38e15·8-s + 5.00e16·9-s + 8.12e16·10-s − 1.47e18·11-s − 1.01e18·12-s + 3.00e19·13-s + 7.31e19·14-s − 3.44e20·15-s − 9.38e20·16-s + 6.18e21·17-s − 3.04e21·18-s − 1.60e22·19-s + 5.24e21·20-s − 3.10e23·21-s + 8.98e22·22-s + 4.93e23·23-s − 3.58e23·24-s − 4.39e24·25-s − 1.82e24·26-s + 8.61e24·27-s + 4.72e24·28-s + ⋯
 L(s)  = 1 − 0.328·2-s + 1.15·3-s − 0.114·4-s − 0.781·5-s − 0.379·6-s − 1.95·7-s − 0.217·8-s + 9-s + 0.256·10-s − 0.879·11-s − 0.132·12-s + 0.962·13-s + 0.641·14-s − 0.902·15-s − 0.794·16-s + 1.81·17-s − 0.328·18-s − 0.671·19-s + 0.0894·20-s − 2.25·21-s + 0.289·22-s + 0.730·23-s − 0.251·24-s − 1.51·25-s − 0.316·26-s + 0.769·27-s + 0.223·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+35/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$9$$    =    $$3^{2}$$ Sign: $1$ Motivic weight: $$35$$ Character: induced by $\chi_{3} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 9,\ (\ :35/2, 35/2),\ 1)$$

## Particular Values

 $$L(18)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{37}{2})$$ 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$C_1$ $$( 1 - p^{17} T )^{2}$$
good2$D_{4}$ $$1 + 3807 p^{4} T + 3732131 p^{11} T^{2} + 3807 p^{39} T^{3} + p^{70} T^{4}$$
5$D_{4}$ $$1 + 266755899348 p T +$$$$19\!\cdots\!58$$$$p^{5} T^{2} + 266755899348 p^{36} T^{3} + p^{70} T^{4}$$
7$D_{4}$ $$1 + 24520669039856 p^{2} T +$$$$46\!\cdots\!86$$$$p^{4} T^{2} + 24520669039856 p^{37} T^{3} + p^{70} T^{4}$$
11$D_{4}$ $$1 + 1474443852221320632 T +$$$$44\!\cdots\!54$$$$p^{2} T^{2} + 1474443852221320632 p^{35} T^{3} + p^{70} T^{4}$$
13$D_{4}$ $$1 - 2308093358323513756 p T +$$$$25\!\cdots\!98$$$$p^{3} T^{2} - 2308093358323513756 p^{36} T^{3} + p^{70} T^{4}$$
17$D_{4}$ $$1 -$$$$36\!\cdots\!96$$$$p T +$$$$10\!\cdots\!78$$$$p^{2} T^{2} -$$$$36\!\cdots\!96$$$$p^{36} T^{3} + p^{70} T^{4}$$
19$D_{4}$ $$1 +$$$$16\!\cdots\!64$$$$T +$$$$29\!\cdots\!62$$$$p T^{2} +$$$$16\!\cdots\!64$$$$p^{35} T^{3} + p^{70} T^{4}$$
23$D_{4}$ $$1 -$$$$49\!\cdots\!72$$$$T +$$$$39\!\cdots\!14$$$$T^{2} -$$$$49\!\cdots\!72$$$$p^{35} T^{3} + p^{70} T^{4}$$
29$D_{4}$ $$1 +$$$$78\!\cdots\!48$$$$T +$$$$15\!\cdots\!22$$$$p T^{2} +$$$$78\!\cdots\!48$$$$p^{35} T^{3} + p^{70} T^{4}$$
31$D_{4}$ $$1 +$$$$12\!\cdots\!68$$$$p^{2} T +$$$$35\!\cdots\!82$$$$p^{2} T^{2} +$$$$12\!\cdots\!68$$$$p^{37} T^{3} + p^{70} T^{4}$$
37$D_{4}$ $$1 -$$$$16\!\cdots\!16$$$$T +$$$$20\!\cdots\!06$$$$T^{2} -$$$$16\!\cdots\!16$$$$p^{35} T^{3} + p^{70} T^{4}$$
41$D_{4}$ $$1 +$$$$32\!\cdots\!88$$$$T +$$$$68\!\cdots\!22$$$$T^{2} +$$$$32\!\cdots\!88$$$$p^{35} T^{3} + p^{70} T^{4}$$
43$D_{4}$ $$1 -$$$$10\!\cdots\!20$$$$T -$$$$10\!\cdots\!10$$$$T^{2} -$$$$10\!\cdots\!20$$$$p^{35} T^{3} + p^{70} T^{4}$$
47$D_{4}$ $$1 +$$$$52\!\cdots\!20$$$$T +$$$$66\!\cdots\!90$$$$T^{2} +$$$$52\!\cdots\!20$$$$p^{35} T^{3} + p^{70} T^{4}$$
53$D_{4}$ $$1 +$$$$31\!\cdots\!68$$$$T -$$$$16\!\cdots\!26$$$$T^{2} +$$$$31\!\cdots\!68$$$$p^{35} T^{3} + p^{70} T^{4}$$
59$D_{4}$ $$1 -$$$$82\!\cdots\!64$$$$T +$$$$20\!\cdots\!58$$$$T^{2} -$$$$82\!\cdots\!64$$$$p^{35} T^{3} + p^{70} T^{4}$$
61$D_{4}$ $$1 +$$$$40\!\cdots\!40$$$$T +$$$$98\!\cdots\!18$$$$T^{2} +$$$$40\!\cdots\!40$$$$p^{35} T^{3} + p^{70} T^{4}$$
67$D_{4}$ $$1 -$$$$96\!\cdots\!12$$$$T +$$$$11\!\cdots\!22$$$$T^{2} -$$$$96\!\cdots\!12$$$$p^{35} T^{3} + p^{70} T^{4}$$
71$D_{4}$ $$1 -$$$$44\!\cdots\!24$$$$T +$$$$99\!\cdots\!46$$$$T^{2} -$$$$44\!\cdots\!24$$$$p^{35} T^{3} + p^{70} T^{4}$$
73$D_{4}$ $$1 +$$$$13\!\cdots\!08$$$$T +$$$$29\!\cdots\!54$$$$T^{2} +$$$$13\!\cdots\!08$$$$p^{35} T^{3} + p^{70} T^{4}$$
79$D_{4}$ $$1 +$$$$10\!\cdots\!80$$$$T +$$$$29\!\cdots\!98$$$$T^{2} +$$$$10\!\cdots\!80$$$$p^{35} T^{3} + p^{70} T^{4}$$
83$D_{4}$ $$1 +$$$$55\!\cdots\!44$$$$T +$$$$36\!\cdots\!82$$$$T^{2} +$$$$55\!\cdots\!44$$$$p^{35} T^{3} + p^{70} T^{4}$$
89$D_{4}$ $$1 -$$$$24\!\cdots\!36$$$$T +$$$$40\!\cdots\!58$$$$T^{2} -$$$$24\!\cdots\!36$$$$p^{35} T^{3} + p^{70} T^{4}$$
97$D_{4}$ $$1 -$$$$83\!\cdots\!52$$$$T +$$$$66\!\cdots\!62$$$$T^{2} -$$$$83\!\cdots\!52$$$$p^{35} T^{3} + p^{70} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$