# Properties

 Degree $4$ Conductor $65536$ Sign $1$ Motivic weight $5$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 416·7-s + 422·9-s − 2.55e3·17-s + 6.43e3·23-s + 6.05e3·25-s + 5.24e3·31-s − 340·41-s − 64·47-s + 9.61e4·49-s − 1.75e5·63-s + 5.71e4·71-s + 1.07e5·73-s + 1.38e5·79-s + 1.19e5·81-s + 2.53e5·89-s + 1.24e5·97-s − 2.16e5·103-s + 3.15e4·113-s + 1.06e6·119-s + 3.48e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.07e6·153-s + ⋯
 L(s)  = 1 − 3.20·7-s + 1.73·9-s − 2.14·17-s + 2.53·23-s + 1.93·25-s + 0.980·31-s − 0.0315·41-s − 0.00422·47-s + 5.72·49-s − 5.57·63-s + 1.34·71-s + 2.35·73-s + 2.49·79-s + 2.01·81-s + 3.39·89-s + 1.34·97-s − 2.01·103-s + 0.232·113-s + 6.88·119-s + 0.216·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 3.72·153-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$65536$$    =    $$2^{16}$$ Sign: $1$ Motivic weight: $$5$$ Character: induced by $\chi_{256} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 65536,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.349282525$$ $$L(\frac12)$$ $$\approx$$ $$2.349282525$$ $$L(\frac{7}{2})$$ 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$C_2^2$ $$1 - 422 T^{2} + p^{10} T^{4}$$
5$C_2^2$ $$1 - 6054 T^{2} + p^{10} T^{4}$$
7$C_2$ $$( 1 + 208 T + p^{5} T^{2} )^{2}$$
11$C_2^2$ $$1 - 34806 T^{2} + p^{10} T^{4}$$
13$C_2^2$ $$1 - 260950 T^{2} + p^{10} T^{4}$$
17$C_2$ $$( 1 + 1278 T + p^{5} T^{2} )^{2}$$
19$C_2^2$ $$1 - 3715654 T^{2} + p^{10} T^{4}$$
23$C_2$ $$( 1 - 3216 T + p^{5} T^{2} )^{2}$$
29$C_2^2$ $$1 - 32507574 T^{2} + p^{10} T^{4}$$
31$C_2$ $$( 1 - 2624 T + p^{5} T^{2} )^{2}$$
37$C_2^2$ $$1 - 49234150 T^{2} + p^{10} T^{4}$$
41$C_2$ $$( 1 + 170 T + p^{5} T^{2} )^{2}$$
43$C_2^2$ $$1 + 103108298 T^{2} + p^{10} T^{4}$$
47$C_2$ $$( 1 + 32 T + p^{5} T^{2} )^{2}$$
53$C_2^2$ $$1 - 344527302 T^{2} + p^{10} T^{4}$$
59$C_2^2$ $$1 + 290741802 T^{2} + p^{10} T^{4}$$
61$C_2^2$ $$1 - 1450119158 T^{2} + p^{10} T^{4}$$
67$C_2^2$ $$1 - 2269936678 T^{2} + p^{10} T^{4}$$
71$C_2$ $$( 1 - 28592 T + p^{5} T^{2} )^{2}$$
73$C_2$ $$( 1 - 53670 T + p^{5} T^{2} )^{2}$$
79$C_2$ $$( 1 - 69152 T + p^{5} T^{2} )^{2}$$
83$C_2^2$ $$1 - 6449241286 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 - 126806 T + p^{5} T^{2} )^{2}$$
97$C_2$ $$( 1 - 62290 T + p^{5} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$