# Properties

 Degree 4 Conductor $5^{2} \cdot 17^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 4·7-s + 8·9-s + 4·10-s − 8·11-s − 4·12-s + 8·14-s + 8·15-s + 16-s − 2·17-s − 16·18-s − 2·20-s + 16·21-s + 16·22-s − 4·23-s + 3·25-s − 12·27-s − 4·28-s − 4·29-s − 16·30-s + 2·32-s + 32·33-s + 4·34-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 8/3·9-s + 1.26·10-s − 2.41·11-s − 1.15·12-s + 2.13·14-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 3.77·18-s − 0.447·20-s + 3.49·21-s + 3.41·22-s − 0.834·23-s + 3/5·25-s − 2.30·27-s − 0.755·28-s − 0.742·29-s − 2.92·30-s + 0.353·32-s + 5.57·33-s + 0.685·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$7225$$    =    $$5^{2} \cdot 17^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7225} (1, \cdot )$ Sato-Tate : $G_{3,3}$ primitive : no self-dual : yes analytic rank = 2 Selberg data = $(4,\ 7225,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;17\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ $$( 1 + T )^{2}$$
17$C_1$ $$( 1 + T )^{2}$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 30 T^{2} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 44 T^{2} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 24 T + 254 T^{2} + 24 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
71$C_2^2$ $$1 + 124 T^{2} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 8 T + 172 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}