# 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 + 3-s + 9·5-s − 2·7-s + 9·9-s + 17·11-s + 9·15-s + 25·17-s − 7·19-s − 2·21-s + 9·23-s + 29·25-s + 26·27-s + 57·31-s + 17·33-s − 18·35-s − 15·37-s + 52·41-s − 28·43-s + 81·45-s − 87·47-s − 45·49-s + 25·51-s − 159·53-s + 153·55-s − 7·57-s − 55·59-s − 39·61-s + ⋯
 L(s)  = 1 + 1/3·3-s + 9/5·5-s − 2/7·7-s + 9-s + 1.54·11-s + 3/5·15-s + 1.47·17-s − 0.368·19-s − 0.0952·21-s + 9/23·23-s + 1.15·25-s + 0.962·27-s + 1.83·31-s + 0.515·33-s − 0.514·35-s − 0.405·37-s + 1.26·41-s − 0.651·43-s + 9/5·45-s − 1.85·47-s − 0.918·49-s + 0.490·51-s − 3·53-s + 2.78·55-s − 0.122·57-s − 0.932·59-s − 0.639·61-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 = $$(4,\ 50176,\ (\ :1, 1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$3.95447$$ $$L(\frac12)$$ $$\approx$$ $$3.95447$$ $$L(2)$$ 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$C_2$ $$1 + 2 T + p^{2} T^{2}$$
good3$C_2^2$ $$1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4}$$
5$C_2^2$ $$1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4}$$
11$C_2^2$ $$1 - 17 T + 168 T^{2} - 17 p^{2} T^{3} + p^{4} T^{4}$$
13$C_2$ $$( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} )$$
17$C_2^2$ $$1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4}$$
19$C_2^2$ $$1 + 7 T - 312 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4}$$
23$C_2^2$ $$1 - 9 T + 556 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4}$$
29$C_2^2$ $$1 - 1490 T^{2} + p^{4} T^{4}$$
31$C_2^2$ $$1 - 57 T + 2044 T^{2} - 57 p^{2} T^{3} + p^{4} T^{4}$$
37$C_2^2$ $$1 + 15 T + 1444 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4}$$
41$C_2$ $$( 1 - 26 T + p^{2} T^{2} )^{2}$$
43$C_2$ $$( 1 + 14 T + p^{2} T^{2} )^{2}$$
47$C_2^2$ $$1 + 87 T + 4732 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4}$$
53$C_1$$\times$$C_2$ $$( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} )$$
59$C_2^2$ $$1 + 55 T - 456 T^{2} + 55 p^{2} T^{3} + p^{4} T^{4}$$
61$C_2^2$ $$1 + 39 T + 4228 T^{2} + 39 p^{2} T^{3} + p^{4} T^{4}$$
67$C_2^2$ $$1 - 17 T - 4200 T^{2} - 17 p^{2} T^{3} + p^{4} T^{4}$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
73$C_2^2$ $$1 + 119 T + 8832 T^{2} + 119 p^{2} T^{3} + p^{4} T^{4}$$
79$C_2^2$ $$1 - 129 T + 11788 T^{2} - 129 p^{2} T^{3} + p^{4} T^{4}$$
83$C_2$ $$( 1 + 110 T + p^{2} T^{2} )^{2}$$
89$C_2^2$ $$1 + 71 T - 2880 T^{2} + 71 p^{2} T^{3} + p^{4} T^{4}$$
97$C_2$ $$( 1 + 22 T + p^{2} T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}