# Properties

 Degree $4$ Conductor $3418801$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·4-s − 2·11-s − 6·13-s − 14·17-s + 2·23-s − 4·25-s − 6·31-s − 10·41-s − 4·44-s − 20·47-s − 8·49-s − 12·52-s − 2·53-s − 20·59-s − 8·64-s + 18·67-s − 28·68-s − 12·79-s − 9·81-s + 2·83-s + 4·92-s + 22·97-s − 8·100-s − 10·101-s + 34·103-s + 28·107-s + 26·109-s + ⋯
 L(s)  = 1 + 4-s − 0.603·11-s − 1.66·13-s − 3.39·17-s + 0.417·23-s − 4/5·25-s − 1.07·31-s − 1.56·41-s − 0.603·44-s − 2.91·47-s − 8/7·49-s − 1.66·52-s − 0.274·53-s − 2.60·59-s − 64-s + 2.19·67-s − 3.39·68-s − 1.35·79-s − 81-s + 0.219·83-s + 0.417·92-s + 2.23·97-s − 4/5·100-s − 0.995·101-s + 3.35·103-s + 2.70·107-s + 2.49·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3418801$$    =    $$43^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1849} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 3418801,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad43 $$1$$
good2$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
3$C_2^2$ $$1 + p^{2} T^{4}$$
5$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
7$C_2^2$ $$1 + 8 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 + T + p T^{2} )^{2}$$
13$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
19$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 - T + p T^{2} )^{2}$$
29$C_2^2$ $$1 + 52 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
37$C_2^2$ $$1 + 50 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 + T + p T^{2} )^{2}$$
59$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
61$C_2^2$ $$1 + 68 T^{2} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
71$C_2^2$ $$1 + 118 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
79$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - 116 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 11 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$