# Properties

 Degree 4 Conductor $2^{4} \cdot 37^{2}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·7-s + 9-s − 2·17-s + 2·19-s − 2·23-s − 2·29-s + 2·47-s + 49-s + 2·53-s − 2·63-s + 2·71-s − 2·79-s − 2·83-s − 2·89-s − 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯
 L(s)  = 1 − 2·7-s + 9-s − 2·17-s + 2·19-s − 2·23-s − 2·29-s + 2·47-s + 49-s + 2·53-s − 2·63-s + 2·71-s − 2·79-s − 2·83-s − 2·89-s − 2·109-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$21904$$    =    $$2^{4} \cdot 37^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{148} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 21904,\ (\ :0, 0),\ 1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.3684437083$$ $$L(\frac12)$$ $$\approx$$ $$0.3684437083$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;37\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
37$C_2$ $$1 + T^{2}$$
good3$C_2^2$ $$1 - T^{2} + T^{4}$$
5$C_2^2$ $$1 + T^{4}$$
7$C_2$ $$( 1 + T + T^{2} )^{2}$$
11$C_2^2$ $$1 - T^{2} + T^{4}$$
13$C_2^2$ $$1 + T^{4}$$
17$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
19$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
29$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
31$C_2^2$ $$1 + T^{4}$$
41$C_2^2$ $$1 - T^{2} + T^{4}$$
43$C_2^2$ $$1 + T^{4}$$
47$C_2$ $$( 1 - T + T^{2} )^{2}$$
53$C_2$ $$( 1 - T + T^{2} )^{2}$$
59$C_2^2$ $$1 + T^{4}$$
61$C_2^2$ $$1 + T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_2$ $$( 1 - T + T^{2} )^{2}$$
73$C_2^2$ $$1 - T^{2} + T^{4}$$
79$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
83$C_2$ $$( 1 + T + T^{2} )^{2}$$
89$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
97$C_2^2$ $$1 + T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}