# Properties

 Degree 4 Conductor $23^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 2·4-s − 2·5-s + 2·7-s + 3·8-s − 9-s + 2·10-s − 6·11-s + 6·13-s − 2·14-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·20-s + 6·22-s + 2·23-s − 2·25-s − 6·26-s − 4·28-s − 6·29-s − 2·32-s − 6·34-s − 4·35-s + 2·36-s + 2·37-s + 4·38-s + ⋯
 L(s)  = 1 − 0.707·2-s − 4-s − 0.894·5-s + 0.755·7-s + 1.06·8-s − 1/3·9-s + 0.632·10-s − 1.80·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.27·22-s + 0.417·23-s − 2/5·25-s − 1.17·26-s − 0.755·28-s − 1.11·29-s − 0.353·32-s − 1.02·34-s − 0.676·35-s + 1/3·36-s + 0.328·37-s + 0.648·38-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$529$$    =    $$23^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{529} (1, \cdot )$ Sato-Tate : $G_{3,3}$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 529,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2484318665$ $L(\frac12)$ $\approx$ $0.2484318665$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 23$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 23$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad23$C_1$ $$( 1 - T )^{2}$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
3$V_4$ $$1 + T^{2} + p^{2} T^{4}$$
5$D_{4}$ $$1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
11$C_4$ $$1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
17$D_{4}$ $$1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
31$V_4$ $$1 + 17 T^{2} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$V_4$ $$1 + 89 T^{2} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 20 T + 237 T^{2} - 20 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 22 T + 247 T^{2} - 22 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4}$$
89$C_4$ $$1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}