# Properties

 Degree 4 Conductor $2^{5} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 8-s − 2·9-s − 8·13-s + 16-s + 12·17-s + 2·18-s − 10·25-s + 8·26-s − 12·29-s − 32-s − 12·34-s − 2·36-s + 4·37-s + 12·41-s + 49-s + 10·50-s − 8·52-s + 12·53-s + 12·58-s + 16·61-s + 64-s + 12·68-s + 2·72-s + 4·73-s − 4·74-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 2.91·17-s + 0.471·18-s − 2·25-s + 1.56·26-s − 2.22·29-s − 0.176·32-s − 2.05·34-s − 1/3·36-s + 0.657·37-s + 1.87·41-s + 1/7·49-s + 1.41·50-s − 1.10·52-s + 1.64·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s + 1.45·68-s + 0.235·72-s + 0.468·73-s − 0.464·74-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1568$$    =    $$2^{5} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1568} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1568,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4377085675$ $L(\frac12)$ $\approx$ $0.4377085675$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

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