Properties

 Label 4-1200e2-1.1-c3e2-0-3 Degree $4$ Conductor $1440000$ Sign $1$ Analytic cond. $5012.96$ Root an. cond. $8.41440$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 9·9-s + 48·11-s − 248·19-s + 156·29-s − 400·31-s + 660·41-s + 286·49-s + 48·59-s − 644·61-s + 576·71-s − 1.04e3·79-s + 81·81-s − 2.05e3·89-s − 432·99-s − 3.46e3·101-s + 2.94e3·109-s − 934·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.08e3·169-s + ⋯
 L(s)  = 1 − 1/3·9-s + 1.31·11-s − 2.99·19-s + 0.998·29-s − 2.31·31-s + 2.51·41-s + 0.833·49-s + 0.105·59-s − 1.35·61-s + 0.962·71-s − 1.48·79-s + 1/9·81-s − 2.44·89-s − 0.438·99-s − 3.41·101-s + 2.59·109-s − 0.701·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.492·169-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$1440000$$    =    $$2^{8} \cdot 3^{2} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$5012.96$$ Root analytic conductor: $$8.41440$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1440000,\ (\ :3/2, 3/2),\ 1)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.7957252257$$ $$L(\frac12)$$ $$\approx$$ $$0.7957252257$$ $$L(\frac{5}{2})$$ 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 $$1$$
3$C_2$ $$1 + p^{2} T^{2}$$
5 $$1$$
good7$C_2^2$ $$1 - 286 T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 - 24 T + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$1 + 1082 T^{2} + p^{6} T^{4}$$
17$C_2^2$ $$1 - 6910 T^{2} + p^{6} T^{4}$$
19$C_2$ $$( 1 + 124 T + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 - 9934 T^{2} + p^{6} T^{4}$$
29$C_2$ $$( 1 - 78 T + p^{3} T^{2} )^{2}$$
31$C_2$ $$( 1 + 200 T + p^{3} T^{2} )^{2}$$
37$C_2^2$ $$1 - 96406 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 - 330 T + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$1 - 150550 T^{2} + p^{6} T^{4}$$
47$C_2^2$ $$1 - 207070 T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 95254 T^{2} + p^{6} T^{4}$$
59$C_2$ $$( 1 - 24 T + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 + 322 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 - 563110 T^{2} + p^{6} T^{4}$$
71$C_2$ $$( 1 - 288 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 - 593134 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 + 520 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 - 1119238 T^{2} + p^{6} T^{4}$$
89$C_2$ $$( 1 + 1026 T + p^{3} T^{2} )^{2}$$
97$C_2^2$ $$1 - 1743550 T^{2} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$