# Properties

 Degree 4 Conductor $2^{4} \cdot 257$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4-s + 3·8-s + 2·9-s + 4·13-s − 16-s + 4·17-s − 2·18-s − 6·25-s − 4·26-s − 12·29-s − 5·32-s − 4·34-s − 2·36-s − 16·37-s + 12·41-s + 2·49-s + 6·50-s − 4·52-s + 8·53-s + 12·58-s + 4·61-s + 7·64-s − 4·68-s + 6·72-s − 12·73-s + 16·74-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.471·18-s − 6/5·25-s − 0.784·26-s − 2.22·29-s − 0.883·32-s − 0.685·34-s − 1/3·36-s − 2.63·37-s + 1.87·41-s + 2/7·49-s + 0.848·50-s − 0.554·52-s + 1.09·53-s + 1.57·58-s + 0.512·61-s + 7/8·64-s − 0.485·68-s + 0.707·72-s − 1.40·73-s + 1.85·74-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$4112$$    =    $$2^{4} \cdot 257$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4112} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 4112,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.5440100461$ $L(\frac12)$ $\approx$ $0.5440100461$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;257\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;257\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T + p T^{2}$$
257$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 2 T + p T^{2} )$$
good3$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
19$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
43$V_4$ $$1 - 18 T^{2} + p^{2} T^{4}$$
47$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
59$V_4$ $$1 + 38 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$V_4$ $$1 + 86 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$V_4$ $$1 - 82 T^{2} + p^{2} T^{4}$$
83$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}