# Properties

 Degree 4 Conductor $2^{4} \cdot 3^{3} \cdot 11$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 3·11-s + 12-s + 4·13-s − 16-s − 18-s − 3·22-s + 8·23-s − 3·24-s − 6·25-s − 4·26-s − 27-s − 5·32-s − 3·33-s − 36-s − 4·37-s − 4·39-s − 3·44-s − 8·46-s − 8·47-s + 48-s + 2·49-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.235·18-s − 0.639·22-s + 1.66·23-s − 0.612·24-s − 6/5·25-s − 0.784·26-s − 0.192·27-s − 0.883·32-s − 0.522·33-s − 1/6·36-s − 0.657·37-s − 0.640·39-s − 0.452·44-s − 1.17·46-s − 1.16·47-s + 0.144·48-s + 2/7·49-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$4752$$    =    $$2^{4} \cdot 3^{3} \cdot 11$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4752} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 4752,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4802185861$ $L(\frac12)$ $\approx$ $0.4802185861$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;11\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, 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$C_1$ $$1 + T$$
11$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 4 T + p T^{2} )$$
good5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - p T^{2} )^{2}$$
19$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$$
29$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 30 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2^2$ $$1 - 114 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - 114 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
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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

−12.22929985017712567797235022800, −11.55281903357586530802457408363, −11.06554417879310638446300217435, −10.59301782729394958636887779974, −9.842386745396970440955645110138, −9.338826184149724184439092868839, −8.770564854925164249035369230573, −8.228260735278396584280170930047, −7.39390228197282041656778383503, −6.74603138048678310026121780637, −5.98467088948894721984616499875, −5.15868189575557544724239962001, −4.31393854123724027348040128745, −3.49049287814637479666504094012, −1.41992576303479051574990026607, 1.41992576303479051574990026607, 3.49049287814637479666504094012, 4.31393854123724027348040128745, 5.15868189575557544724239962001, 5.98467088948894721984616499875, 6.74603138048678310026121780637, 7.39390228197282041656778383503, 8.228260735278396584280170930047, 8.770564854925164249035369230573, 9.338826184149724184439092868839, 9.842386745396970440955645110138, 10.59301782729394958636887779974, 11.06554417879310638446300217435, 11.55281903357586530802457408363, 12.22929985017712567797235022800