# Properties

 Degree 4 Conductor $3^{6} \cdot 5^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4-s − 2·5-s − 2·7-s + 4·10-s + 2·11-s + 2·13-s + 4·14-s + 16-s − 4·17-s + 6·19-s − 2·20-s − 4·22-s − 4·23-s + 3·25-s − 4·26-s − 2·28-s + 8·29-s + 12·31-s + 2·32-s + 8·34-s + 4·35-s − 12·38-s + 10·41-s − 2·43-s + 2·44-s + 8·46-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 1.26·10-s + 0.603·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.447·20-s − 0.852·22-s − 0.834·23-s + 3/5·25-s − 0.784·26-s − 0.377·28-s + 1.48·29-s + 2.15·31-s + 0.353·32-s + 1.37·34-s + 0.676·35-s − 1.94·38-s + 1.56·41-s − 0.304·43-s + 0.301·44-s + 1.17·46-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$893025$$    =    $$3^{6} \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{945} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 893025,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.684654$ $L(\frac12)$ $\approx$ $0.684654$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;7\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 $$1$$
5$C_1$ $$( 1 + T )^{2}$$
7$C_1$ $$( 1 + T )^{2}$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
23$C_4$ $$1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
37$V_4$ $$1 + 2 T^{2} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
61$V_4$ $$1 + 50 T^{2} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 10 T + 151 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
73$D_{4}$ $$1 - 10 T + 121 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 2 T + 95 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
97$V_4$ $$1 - 6 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}