# Properties

 Degree 4 Conductor $2^{2} \cdot 5^{2} \cdot 17^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·3-s − 4-s − 4·5-s − 4·7-s − 3·9-s + 2·12-s + 8·15-s + 16-s − 2·17-s − 10·19-s + 4·20-s + 8·21-s + 8·23-s + 11·25-s + 14·27-s + 4·28-s + 16·35-s + 3·36-s − 4·37-s + 12·45-s − 2·48-s − 2·49-s + 4·51-s + 20·57-s − 10·59-s − 8·60-s + 12·63-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1/2·4-s − 1.78·5-s − 1.51·7-s − 9-s + 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 2.29·19-s + 0.894·20-s + 1.74·21-s + 1.66·23-s + 11/5·25-s + 2.69·27-s + 0.755·28-s + 2.70·35-s + 1/2·36-s − 0.657·37-s + 1.78·45-s − 0.288·48-s − 2/7·49-s + 0.560·51-s + 2.64·57-s − 1.30·59-s − 1.03·60-s + 1.51·63-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$28900$$    =    $$2^{2} \cdot 5^{2} \cdot 17^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{28900} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 28900,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.08351315462$ $L(\frac12)$ $\approx$ $0.08351315462$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;17\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T^{2}$$
5$C_2$ $$1 + 4 T + p T^{2}$$
17$C_2$ $$1 + 2 T + p T^{2}$$
good3$C_2$ $$( 1 + T + p T^{2} )^{2}$$
7$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - p T^{2} )^{2}$$
13$C_2^2$ $$1 - 25 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
29$C_2^2$ $$1 + 23 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 37 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 - 45 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 105 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
61$C_2^2$ $$1 - 97 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 117 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 + 11 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 + 98 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 130 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}