# Properties

 Degree 4 Conductor $2^{10} \cdot 7^{2}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 14·7-s − 10·9-s + 20·23-s + 22·25-s + 147·49-s − 140·63-s + 220·71-s − 260·79-s + 19·81-s − 52·113-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 280·161-s + 163-s + 167-s + 310·169-s + 173-s + 308·175-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 + 2·7-s − 1.11·9-s + 0.869·23-s + 0.879·25-s + 3·49-s − 2.22·63-s + 3.09·71-s − 3.29·79-s + 0.234·81-s − 0.460·113-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 1.73·161-s + 0.00613·163-s + 0.00598·167-s + 1.83·169-s + 0.00578·173-s + 1.75·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$50176$$    =    $$2^{10} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{224} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 50176,\ (\ :1, 1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.38121$$ $$L(\frac12)$$ $$\approx$$ $$2.38121$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
7$C_1$ $$( 1 - p T )^{2}$$
good3$C_2^2$ $$1 + 10 T^{2} + p^{4} T^{4}$$
5$C_2^2$ $$1 - 22 T^{2} + p^{4} T^{4}$$
11$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
13$C_2^2$ $$1 - 310 T^{2} + p^{4} T^{4}$$
17$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
19$C_2^2$ $$1 + 650 T^{2} + p^{4} T^{4}$$
23$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
37$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
43$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
53$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
59$C_2^2$ $$1 + 1130 T^{2} + p^{4} T^{4}$$
61$C_2^2$ $$1 + 7370 T^{2} + p^{4} T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
71$C_2$ $$( 1 - 110 T + p^{2} T^{2} )^{2}$$
73$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
79$C_2$ $$( 1 + 130 T + p^{2} T^{2} )^{2}$$
83$C_2^2$ $$1 + 13130 T^{2} + p^{4} T^{4}$$
89$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}