# Properties

 Degree 4 Conductor $2^{4} \cdot 5^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 8·4-s − 4·5-s − 16·10-s − 74·13-s − 64·16-s + 198·17-s − 32·20-s − 109·25-s − 296·26-s − 256·32-s + 792·34-s − 182·37-s + 944·41-s − 436·50-s − 592·52-s − 54·53-s − 936·61-s − 512·64-s + 296·65-s + 1.58e3·68-s + 506·73-s − 728·74-s + 256·80-s − 729·81-s + 3.77e3·82-s − 792·85-s + ⋯
 L(s)  = 1 + 1.41·2-s + 4-s − 0.357·5-s − 0.505·10-s − 1.57·13-s − 16-s + 2.82·17-s − 0.357·20-s − 0.871·25-s − 2.23·26-s − 1.41·32-s + 3.99·34-s − 0.808·37-s + 3.59·41-s − 1.23·50-s − 1.57·52-s − 0.139·53-s − 1.96·61-s − 64-s + 0.564·65-s + 2.82·68-s + 0.811·73-s − 1.14·74-s + 0.357·80-s − 81-s + 5.08·82-s − 1.01·85-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$400$$    =    $$2^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{20} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 400,\ (\ :3/2, 3/2),\ 1)$ $L(2)$ $\approx$ $1.96950$ $L(\frac12)$ $\approx$ $1.96950$ $L(\frac{5}{2})$ 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$C_2$ $$1 - p^{2} T + p^{3} T^{2}$$
5$C_2$ $$1 + 4 T + p^{3} T^{2}$$
good3$C_2^2$ $$1 + p^{6} T^{4}$$
7$C_2^2$ $$1 + p^{6} T^{4}$$
11$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
13$C_2$ $$( 1 - 18 T + p^{3} T^{2} )( 1 + 92 T + p^{3} T^{2} )$$
17$C_2$ $$( 1 - 104 T + p^{3} T^{2} )( 1 - 94 T + p^{3} T^{2} )$$
19$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 + p^{6} T^{4}$$
29$C_2$ $$( 1 - 130 T + p^{3} T^{2} )( 1 + 130 T + p^{3} T^{2} )$$
31$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
37$C_2$ $$( 1 - 214 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} )$$
41$C_2$ $$( 1 - 472 T + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$1 + p^{6} T^{4}$$
47$C_2^2$ $$1 + p^{6} T^{4}$$
53$C_2$ $$( 1 - 518 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} )$$
59$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 + 468 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 + p^{6} T^{4}$$
71$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
73$C_2$ $$( 1 - 1098 T + p^{3} T^{2} )( 1 + 592 T + p^{3} T^{2} )$$
79$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 + p^{6} T^{4}$$
89$C_2$ $$( 1 - 1670 T + p^{3} T^{2} )( 1 + 1670 T + p^{3} T^{2} )$$
97$C_2$ $$( 1 - 594 T + p^{3} T^{2} )( 1 + 1816 T + p^{3} T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}