# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 11-s − 4·16-s − 11·31-s + 9·41-s − 61-s + 19·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 0.301·11-s − 16-s − 1.97·31-s + 1.40·41-s − 0.128·61-s + 2.25·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.301·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$3125$$    =    $$5^{5}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{3125} (1, \cdot )$ Sato-Tate : $F_{ac}$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 3125,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.7183136284$ $L(\frac12)$ $\approx$ $0.7183136284$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5 $$1$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
3$V_4$ $$1 + p^{2} T^{4}$$
7$V_4$ $$1 + p^{2} T^{4}$$
11$C_4$ $$1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4}$$
13$V_4$ $$1 + p^{2} T^{4}$$
17$V_4$ $$1 + p^{2} T^{4}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$V_4$ $$1 + p^{2} T^{4}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_4$ $$1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
37$V_4$ $$1 + p^{2} T^{4}$$
41$C_4$ $$1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
43$V_4$ $$1 + p^{2} T^{4}$$
47$V_4$ $$1 + p^{2} T^{4}$$
53$V_4$ $$1 + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_4$ $$1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4}$$
67$V_4$ $$1 + p^{2} T^{4}$$
71$C_4$ $$1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4}$$
73$V_4$ $$1 + p^{2} T^{4}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$V_4$ $$1 + p^{2} T^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$V_4$ $$1 + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}