# Properties

 Label 4-33e4-1.1-c3e2-0-9 Degree $4$ Conductor $1185921$ Sign $1$ Analytic cond. $4128.45$ Root an. cond. $8.01580$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 11·4-s − 18·5-s + 57·16-s − 198·20-s − 180·23-s − 7·25-s − 376·31-s + 266·37-s − 144·47-s − 98·49-s + 90·53-s − 756·59-s − 77·64-s − 772·67-s + 396·71-s − 1.02e3·80-s − 90·89-s − 1.98e3·92-s + 178·97-s − 77·100-s + 2.71e3·103-s − 1.42e3·113-s + 3.24e3·115-s − 4.13e3·124-s + 3.83e3·125-s + 127-s + 131-s + ⋯
 L(s)  = 1 + 11/8·4-s − 1.60·5-s + 0.890·16-s − 2.21·20-s − 1.63·23-s − 0.0559·25-s − 2.17·31-s + 1.18·37-s − 0.446·47-s − 2/7·49-s + 0.233·53-s − 1.66·59-s − 0.150·64-s − 1.40·67-s + 0.661·71-s − 1.43·80-s − 0.107·89-s − 2.24·92-s + 0.186·97-s − 0.0769·100-s + 2.59·103-s − 1.18·113-s + 2.62·115-s − 2.99·124-s + 2.74·125-s + 0.000698·127-s + 0.000666·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1185921$$    =    $$3^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$4128.45$$ Root analytic conductor: $$8.01580$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1185921,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 $$1$$
11 $$1$$
good2$C_2^2$ $$1 - 11 T^{2} + p^{6} T^{4}$$
5$C_2$ $$( 1 + 9 T + p^{3} T^{2} )^{2}$$
7$C_2^2$ $$1 + 2 p^{2} T^{2} + p^{6} T^{4}$$
13$C_2^2$ $$1 - 649 T^{2} + p^{6} T^{4}$$
17$C_2^2$ $$1 + 7 p^{2} T^{2} + p^{6} T^{4}$$
19$C_2^2$ $$1 - 7450 T^{2} + p^{6} T^{4}$$
23$C_2$ $$( 1 + 90 T + p^{3} T^{2} )^{2}$$
29$C_2^2$ $$1 + 40975 T^{2} + p^{6} T^{4}$$
31$C_2$ $$( 1 + 188 T + p^{3} T^{2} )^{2}$$
37$C_2$ $$( 1 - 133 T + p^{3} T^{2} )^{2}$$
41$C_2^2$ $$1 + 136519 T^{2} + p^{6} T^{4}$$
43$C_2^2$ $$1 + 153722 T^{2} + p^{6} T^{4}$$
47$C_2$ $$( 1 + 72 T + p^{3} T^{2} )^{2}$$
53$C_2$ $$( 1 - 45 T + p^{3} T^{2} )^{2}$$
59$C_2$ $$( 1 + 378 T + p^{3} T^{2} )^{2}$$
61$C_2^2$ $$1 + 65162 T^{2} + p^{6} T^{4}$$
67$C_2$ $$( 1 + 386 T + p^{3} T^{2} )^{2}$$
71$C_2$ $$( 1 - 198 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 772226 T^{2} + p^{6} T^{4}$$
79$C_2^2$ $$1 + 962846 T^{2} + p^{6} T^{4}$$
83$C_2^2$ $$1 - 411626 T^{2} + p^{6} T^{4}$$
89$C_2$ $$( 1 + 45 T + p^{3} T^{2} )^{2}$$
97$C_2$ $$( 1 - 89 T + p^{3} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$