# Properties

 Label 4-18e2-1.1-c8e2-0-0 Degree $4$ Conductor $324$ Sign $1$ Analytic cond. $53.7701$ Root an. cond. $2.70791$ Motivic weight $8$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 128·4-s − 7.06e3·7-s − 8.36e4·13-s + 1.63e4·16-s − 7.26e4·19-s + 7.26e5·25-s + 9.04e5·28-s − 9.42e5·31-s − 6.01e6·37-s + 7.24e6·43-s + 2.58e7·49-s + 1.07e7·52-s − 1.08e7·61-s − 2.09e6·64-s − 1.22e7·67-s − 9.80e7·73-s + 9.29e6·76-s + 1.67e7·79-s + 5.90e8·91-s + 4.08e7·97-s − 9.30e7·100-s − 5.96e7·103-s − 9.77e7·109-s − 1.15e8·112-s + 2.15e7·121-s + 1.20e8·124-s + 127-s + ⋯
 L(s)  = 1 − 1/2·4-s − 2.94·7-s − 2.92·13-s + 1/4·16-s − 0.557·19-s + 1.86·25-s + 1.47·28-s − 1.02·31-s − 3.20·37-s + 2.11·43-s + 4.49·49-s + 1.46·52-s − 0.785·61-s − 1/8·64-s − 0.607·67-s − 3.45·73-s + 0.278·76-s + 0.429·79-s + 8.61·91-s + 0.461·97-s − 0.930·100-s − 0.529·103-s − 0.692·109-s − 0.735·112-s + 0.100·121-s + 0.510·124-s + 1.63·133-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$53.7701$$ Root analytic conductor: $$2.70791$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 324,\ (\ :4, 4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.03803402313$$ $$L(\frac12)$$ $$\approx$$ $$0.03803402313$$ $$L(5)$$ 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)$
bad2$C_2$ $$1 + p^{7} T^{2}$$
3 $$1$$
good5$C_2^2$ $$1 - 29072 p^{2} T^{2} + p^{16} T^{4}$$
7$C_2$ $$( 1 + 3532 T + p^{8} T^{2} )^{2}$$
11$C_2^2$ $$1 - 21566114 T^{2} + p^{16} T^{4}$$
13$C_2$ $$( 1 + 41824 T + p^{8} T^{2} )^{2}$$
17$C_2^2$ $$1 - 4967349824 T^{2} + p^{16} T^{4}$$
19$C_2$ $$( 1 + 36304 T + p^{8} T^{2} )^{2}$$
23$C_2^2$ $$1 + 14462450590 T^{2} + p^{16} T^{4}$$
29$C_2^2$ $$1 - 928028865104 T^{2} + p^{16} T^{4}$$
31$C_2$ $$( 1 + 471196 T + p^{8} T^{2} )^{2}$$
37$C_2$ $$( 1 + 3007402 T + p^{8} T^{2} )^{2}$$
41$C_2^2$ $$1 - 13027466643584 T^{2} + p^{16} T^{4}$$
43$C_2$ $$( 1 - 3623720 T + p^{8} T^{2} )^{2}$$
47$C_2^2$ $$1 - 11446895562722 T^{2} + p^{16} T^{4}$$
53$C_2^2$ $$1 - 18909552109520 T^{2} + p^{16} T^{4}$$
59$C_2^2$ $$1 - 286434976404290 T^{2} + p^{16} T^{4}$$
61$C_2$ $$( 1 + 5440630 T + p^{8} T^{2} )^{2}$$
67$C_2$ $$( 1 + 6121576 T + p^{8} T^{2} )^{2}$$
71$C_2^2$ $$1 - 842318136694370 T^{2} + p^{16} T^{4}$$
73$C_2$ $$( 1 + 49031152 T + p^{8} T^{2} )^{2}$$
79$C_2$ $$( 1 - 8357756 T + p^{8} T^{2} )^{2}$$
83$C_2^2$ $$1 - 1863467106641954 T^{2} + p^{16} T^{4}$$
89$C_2^2$ $$1 + 3648102662700160 T^{2} + p^{16} T^{4}$$
97$C_2$ $$( 1 - 20431328 T + p^{8} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$