# Properties

 Degree 4 Conductor $2^{2} \cdot 17^{3}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 4·13-s + 5·16-s − 17-s − 4·18-s − 8·19-s − 10·25-s + 8·26-s + 6·32-s − 2·34-s − 6·36-s − 16·38-s + 16·43-s + 2·49-s − 20·50-s + 12·52-s − 12·53-s + 7·64-s + 16·67-s − 3·68-s − 8·72-s − 24·76-s − 5·81-s + 32·86-s + ⋯
 L(s)  = 1 + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2/3·9-s + 1.10·13-s + 5/4·16-s − 0.242·17-s − 0.942·18-s − 1.83·19-s − 2·25-s + 1.56·26-s + 1.06·32-s − 0.342·34-s − 36-s − 2.59·38-s + 2.43·43-s + 2/7·49-s − 2.82·50-s + 1.66·52-s − 1.64·53-s + 7/8·64-s + 1.95·67-s − 0.363·68-s − 0.942·72-s − 2.75·76-s − 5/9·81-s + 3.45·86-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$19652$$    =    $$2^{2} \cdot 17^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{19652} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 19652,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$2.450942393$$ $$L(\frac12)$$ $$\approx$$ $$2.450942393$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

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