# Properties

 Degree 4 Conductor $2^{4} \cdot 5^{2}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s − 6·5-s − 14·7-s + 2·9-s + 20·11-s + 18·13-s − 12·15-s + 2·17-s − 28·21-s − 46·23-s + 11·25-s + 18·27-s − 28·31-s + 40·33-s + 84·35-s + 66·37-s + 36·39-s − 28·41-s − 30·43-s − 12·45-s − 78·47-s + 98·49-s + 4·51-s − 14·53-s − 120·55-s + 84·61-s − 28·63-s + ⋯
 L(s)  = 1 + 2/3·3-s − 6/5·5-s − 2·7-s + 2/9·9-s + 1.81·11-s + 1.38·13-s − 4/5·15-s + 2/17·17-s − 4/3·21-s − 2·23-s + 0.439·25-s + 2/3·27-s − 0.903·31-s + 1.21·33-s + 12/5·35-s + 1.78·37-s + 0.923·39-s − 0.682·41-s − 0.697·43-s − 0.266·45-s − 1.65·47-s + 2·49-s + 4/51·51-s − 0.264·53-s − 2.18·55-s + 1.37·61-s − 4/9·63-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$400$$    =    $$2^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{20} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 400,\ (\ :1, 1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.733358$ $L(\frac12)$ $\approx$ $0.733358$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
5$C_2$ $$1 + 6 T + p^{2} T^{2}$$
good3$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4}$$
7$C_1$$\times$$C_2$ $$( 1 + p T )^{2}( 1 + p^{2} T^{2} )$$
11$C_2$ $$( 1 - 10 T + p^{2} T^{2} )^{2}$$
13$C_2^2$ $$1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4}$$
17$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4}$$
19$C_2^2$ $$1 - 658 T^{2} + p^{4} T^{4}$$
23$C_1$$\times$$C_2$ $$( 1 + p T )^{2}( 1 + p^{2} T^{2} )$$
29$C_2^2$ $$1 - 1618 T^{2} + p^{4} T^{4}$$
31$C_2$ $$( 1 + 14 T + p^{2} T^{2} )^{2}$$
37$C_2^2$ $$1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4}$$
41$C_2$ $$( 1 + 14 T + p^{2} T^{2} )^{2}$$
43$C_2^2$ $$1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4}$$
47$C_2^2$ $$1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4}$$
53$C_2^2$ $$1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4}$$
59$C_2^2$ $$1 - 3826 T^{2} + p^{4} T^{4}$$
61$C_2$ $$( 1 - 42 T + p^{2} T^{2} )^{2}$$
67$C_2^2$ $$1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4}$$
71$C_2$ $$( 1 - 98 T + p^{2} T^{2} )^{2}$$
73$C_2^2$ $$1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4}$$
79$C_2^2$ $$1 - 3266 T^{2} + p^{4} T^{4}$$
83$C_2^2$ $$1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4}$$
89$C_2^2$ $$1 - 3298 T^{2} + p^{4} T^{4}$$
97$C_2^2$ $$1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}