# Properties

 Degree $4$ Conductor $5308416$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·11-s − 4·13-s + 8·23-s + 8·25-s − 16·37-s − 24·47-s + 6·49-s − 16·61-s + 8·71-s − 16·73-s + 24·83-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 − 2.41·11-s − 1.10·13-s + 1.66·23-s + 8/5·25-s − 2.63·37-s − 3.50·47-s + 6/7·49-s − 2.04·61-s + 0.949·71-s − 1.87·73-s + 2.63·83-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$5308416$$    =    $$2^{16} \cdot 3^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{2304} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 5308416,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4617232442$$ $$L(\frac12)$$ $$\approx$$ $$0.4617232442$$ $$L(\frac{3}{2})$$ 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)$
bad2 $$1$$
3 $$1$$
good5$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
7$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 - 32 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
29$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 64 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 42 T^{2} + p^{2} T^{4}$$
47$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
53$C_2^2$ $$1 + 56 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
67$C_2^2$ $$1 - 102 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - 150 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 + 64 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$