# Properties

 Degree 4 Conductor $2^{2} \cdot 5^{5}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s + 4·11-s + 16-s + 5·19-s + 5·29-s − 31-s − 41-s − 4·44-s − 10·49-s − 15·59-s − 6·61-s − 64-s − 11·71-s − 5·76-s + 10·79-s − 9·81-s + 20·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 5·116-s − 10·121-s + 124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 1/2·4-s + 1.20·11-s + 1/4·16-s + 1.14·19-s + 0.928·29-s − 0.179·31-s − 0.156·41-s − 0.603·44-s − 1.42·49-s − 1.95·59-s − 0.768·61-s − 1/8·64-s − 1.30·71-s − 0.573·76-s + 1.12·79-s − 81-s + 2.11·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.464·116-s − 0.909·121-s + 0.0898·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$12500$$    =    $$2^{2} \cdot 5^{5}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{12500} (1, \cdot )$ Sato-Tate : $N(G_{3,3})$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 12500,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.000455522$ $L(\frac12)$ $\approx$ $1.000455522$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T^{2}$$
5 $$1$$
good3$V_4$ $$1 + p^{2} T^{4}$$
7$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
13$V_4$ $$1 + p^{2} T^{4}$$
17$V_4$ $$1 - 20 T^{2} + p^{2} T^{4}$$
19$D_4$ $$1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
23$V_4$ $$1 - 30 T^{2} + p^{2} T^{4}$$
29$D_4$ $$1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
31$D_4$ $$1 + T + p T^{2} + p T^{3} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$D_4$ $$1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$V_4$ $$1 + 70 T^{2} + p^{2} T^{4}$$
53$V_4$ $$1 + 40 T^{2} + p^{2} T^{4}$$
59$D_4$ $$1 + 15 T + 143 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
61$D_4$ $$1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
67$V_4$ $$1 - 80 T^{2} + p^{2} T^{4}$$
71$D_4$ $$1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$D_4$ $$1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
83$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
97$V_4$ $$1 - 140 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}