# Properties

 Degree 4 Conductor 68209 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 4-s − 5·5-s + 6·6-s − 5·7-s + 2·9-s + 10·10-s − 6·11-s − 3·12-s − 4·13-s + 10·14-s + 15·15-s + 16-s − 3·17-s − 4·18-s − 4·19-s − 5·20-s + 15·21-s + 12·22-s + 12·25-s + 8·26-s + 6·27-s − 5·28-s − 2·29-s − 30·30-s − 8·31-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 2/3·9-s + 3.16·10-s − 1.80·11-s − 0.866·12-s − 1.10·13-s + 2.67·14-s + 3.87·15-s + 1/4·16-s − 0.727·17-s − 0.942·18-s − 0.917·19-s − 1.11·20-s + 3.27·21-s + 2.55·22-s + 12/5·25-s + 1.56·26-s + 1.15·27-s − 0.944·28-s − 0.371·29-s − 5.47·30-s − 1.43·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 68209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 68209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$68209$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{68209} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(4,\ 68209,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 68209$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 68209$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad68209$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 164 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$D_{4}$ $$1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
19$D_{4}$ $$1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
23$C_2^2$ $$1 + 14 T^{2} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$D_{4}$ $$1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 13 T + 131 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$D_{4}$ $$1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
97$C_2^2$ $$1 + 110 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}