# Properties

 Degree 4 Conductor $2^{6} \cdot 3^{2} \cdot 17^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 4-s − 3·8-s + 9-s − 16-s − 2·17-s + 18-s − 10·25-s + 5·32-s − 2·34-s − 36-s − 16·47-s − 2·49-s − 10·50-s + 7·64-s + 2·68-s − 3·72-s + 81-s − 20·89-s − 16·94-s − 2·98-s + 10·100-s − 6·121-s + 127-s − 3·128-s + 131-s + 6·136-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 2·25-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 2.33·47-s − 2/7·49-s − 1.41·50-s + 7/8·64-s + 0.242·68-s − 0.353·72-s + 1/9·81-s − 2.11·89-s − 1.65·94-s − 0.202·98-s + 100-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.514·136-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$166464$$    =    $$2^{6} \cdot 3^{2} \cdot 17^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{166464} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 166464,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;17\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 - T + p T^{2}$$
3$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
17$C_2$ $$1 + 2 T + p T^{2}$$
good5$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2^2$ $$1 - 46 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - p T^{2} )^{2}$$
79$C_2^2$ $$1 - 142 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
97$C_2^2$ $$1 + 62 T^{2} + p^{2} T^{4}$$
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\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}

## Imaginary part of the first few zeros on the critical line

−8.990627621060578483098017594856, −8.466527665107842666653023212386, −8.040243271667994896507365067817, −7.61015041301891599610023453537, −6.85505714070999859977992001649, −6.45990116372875969103688741620, −5.87647456361852564136261392180, −5.45184408676781858561883617452, −4.81099171488550293269767138312, −4.36159397767822262013932326900, −3.79884678729776031688190779911, −3.28532150713041810310982697520, −2.45336032252096177013300497760, −1.56769530060653144854680691564, 0, 1.56769530060653144854680691564, 2.45336032252096177013300497760, 3.28532150713041810310982697520, 3.79884678729776031688190779911, 4.36159397767822262013932326900, 4.81099171488550293269767138312, 5.45184408676781858561883617452, 5.87647456361852564136261392180, 6.45990116372875969103688741620, 6.85505714070999859977992001649, 7.61015041301891599610023453537, 8.040243271667994896507365067817, 8.466527665107842666653023212386, 8.990627621060578483098017594856