Properties

 Label 4-60e3-1.1-c1e2-0-16 Degree $4$ Conductor $216000$ Sign $-1$ Analytic cond. $13.7723$ Root an. cond. $1.92642$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s + 12-s − 4·13-s − 15-s − 16-s + 4·17-s − 18-s + 8·19-s − 20-s − 3·24-s + 25-s + 4·26-s − 27-s − 4·29-s + 30-s − 5·32-s − 4·34-s − 36-s − 20·37-s − 8·38-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.742·29-s + 0.182·30-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s − 1.29·38-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$216000$$    =    $$2^{6} \cdot 3^{3} \cdot 5^{3}$$ Sign: $-1$ Analytic conductor: $$13.7723$$ Root analytic conductor: $$1.92642$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 216000,\ (\ :1/2, 1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_2$ $$1 + T + p T^{2}$$
3$C_1$ $$1 + T$$
5$C_1$ $$1 - T$$
good7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
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$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$