# Properties

 Degree $4$ Conductor $65536$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·7-s + 2·9-s − 4·17-s + 8·23-s + 6·25-s + 12·41-s + 16·47-s + 34·49-s − 16·63-s + 24·71-s − 28·73-s + 16·79-s − 5·81-s + 4·89-s − 4·97-s − 8·103-s + 4·113-s + 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯
 L(s)  = 1 − 3.02·7-s + 2/3·9-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 1.87·41-s + 2.33·47-s + 34/7·49-s − 2.01·63-s + 2.84·71-s − 3.27·73-s + 1.80·79-s − 5/9·81-s + 0.423·89-s − 0.406·97-s − 0.788·103-s + 0.376·113-s + 2.93·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$65536$$    =    $$2^{16}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{256} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 65536,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.943494$$ $$L(\frac12)$$ $$\approx$$ $$0.943494$$ $$L(\frac{3}{2})$$ 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 $$1$$
good3$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
11$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
13$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2^2$ $$1 - 34 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
43$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
53$C_2^2$ $$1 - 70 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 78 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 118 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 34 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
83$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$