# Properties

 Degree $4$ Conductor $1539$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s − 2·7-s − 2·13-s − 3·16-s + 3·19-s − 4·25-s + 2·28-s + 10·31-s + 4·37-s + 4·43-s − 2·49-s + 2·52-s − 8·61-s + 7·64-s − 2·67-s + 4·73-s − 3·76-s + 4·79-s + 4·91-s − 14·97-s + 4·100-s − 8·103-s − 14·109-s + 6·112-s + 14·121-s − 10·124-s + 127-s + ⋯
 L(s)  = 1 − 1/2·4-s − 0.755·7-s − 0.554·13-s − 3/4·16-s + 0.688·19-s − 4/5·25-s + 0.377·28-s + 1.79·31-s + 0.657·37-s + 0.609·43-s − 2/7·49-s + 0.277·52-s − 1.02·61-s + 7/8·64-s − 0.244·67-s + 0.468·73-s − 0.344·76-s + 0.450·79-s + 0.419·91-s − 1.42·97-s + 2/5·100-s − 0.788·103-s − 1.34·109-s + 0.566·112-s + 1.27·121-s − 0.898·124-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1539$$    =    $$3^{4} \cdot 19$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{1539} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1539,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5221435606$$ $$L(\frac12)$$ $$\approx$$ $$0.5221435606$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
19$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 2 T + p T^{2} )$$
good2$C_2^2$ $$1 + T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2^2$ $$1 + 70 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
59$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2^2$ $$1 - 86 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−13.68816362801267509972038229581, −13.15510058006465130171422176517, −12.40095477290838686937624430215, −11.89355091746708574088114751364, −11.20226839691820943158562643086, −10.34859675467964656496681254428, −9.600392817368737572042804110266, −9.384260671410315056207093168535, −8.389258775251883182185131053515, −7.66716851151729354484994997123, −6.78439873765208802194785716556, −6.03804535320455140917727072081, −4.98638130370676266639532417421, −4.09729391479811292149019045922, −2.78666419482573957815444693787, 2.78666419482573957815444693787, 4.09729391479811292149019045922, 4.98638130370676266639532417421, 6.03804535320455140917727072081, 6.78439873765208802194785716556, 7.66716851151729354484994997123, 8.389258775251883182185131053515, 9.384260671410315056207093168535, 9.600392817368737572042804110266, 10.34859675467964656496681254428, 11.20226839691820943158562643086, 11.89355091746708574088114751364, 12.40095477290838686937624430215, 13.15510058006465130171422176517, 13.68816362801267509972038229581