# Properties

 Degree 4 Conductor $5^{2} \cdot 61^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 2

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·4-s − 3·5-s − 2·9-s − 10·11-s + 5·16-s − 8·19-s + 9·20-s + 4·25-s − 12·29-s + 6·36-s + 10·41-s + 30·44-s + 6·45-s − 13·49-s + 30·55-s + 18·59-s − 2·61-s − 3·64-s − 16·71-s + 24·76-s + 6·79-s − 15·80-s − 5·81-s − 8·89-s + 24·95-s + 20·99-s − 12·100-s + ⋯
 L(s)  = 1 − 3/2·4-s − 1.34·5-s − 2/3·9-s − 3.01·11-s + 5/4·16-s − 1.83·19-s + 2.01·20-s + 4/5·25-s − 2.22·29-s + 36-s + 1.56·41-s + 4.52·44-s + 0.894·45-s − 1.85·49-s + 4.04·55-s + 2.34·59-s − 0.256·61-s − 3/8·64-s − 1.89·71-s + 2.75·76-s + 0.675·79-s − 1.67·80-s − 5/9·81-s − 0.847·89-s + 2.46·95-s + 2.01·99-s − 6/5·100-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$93025$$    =    $$5^{2} \cdot 61^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{93025} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 2 Selberg data = $(4,\ 93025,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;61\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ $$1 + 3 T + p T^{2}$$
61$C_1$ $$( 1 + T )^{2}$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
17$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
79$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
89$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}