# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 6·7-s − 3·8-s + 6·9-s + 9·10-s − 16·12-s + 2·13-s + 18·14-s + 12·15-s + 3·16-s − 12·17-s − 18·18-s + 2·19-s − 12·20-s + 24·21-s + 12·24-s + 5·25-s − 6·26-s + 4·27-s − 24·28-s − 3·29-s − 36·30-s − 18·31-s + ⋯
 L(s)  = 1 − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 2.26·7-s − 1.06·8-s + 2·9-s + 2.84·10-s − 4.61·12-s + 0.554·13-s + 4.81·14-s + 3.09·15-s + 3/4·16-s − 2.91·17-s − 4.24·18-s + 0.458·19-s − 2.68·20-s + 5.23·21-s + 2.44·24-s + 25-s − 1.17·26-s + 0.769·27-s − 4.53·28-s − 0.557·29-s − 6.57·30-s − 3.23·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$3721$$    =    $$61^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{3721} (1, \cdot )$ Sato-Tate : $E_6$ primitive : no self-dual : yes analytic rank = 2 Selberg data = $(4,\ 3721,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 61$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 61$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad61$C_2$ $$1 + T + p T^{2}$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
3$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
5$C_2^2$ $$1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
17$C_2^2$ $$1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
23$C_2$ $$( 1 - p T^{2} )^{2}$$
29$C_2^2$ $$1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
31$C_2$ $$( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
37$C_2^2$ $$1 - 71 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
43$C_2^2$ $$1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 79 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
71$C_2^2$ $$1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2^2$ $$1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 65 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}