# Properties

 Label 4-15e4-1.1-c3e2-0-8 Degree $4$ Conductor $50625$ Sign $1$ Analytic cond. $176.237$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·4-s + 192·16-s − 112·19-s + 616·31-s + 286·49-s + 364·61-s + 2.04e3·64-s − 1.79e3·76-s − 1.76e3·79-s + 1.29e3·109-s − 2.66e3·121-s + 9.85e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 2·4-s + 3·16-s − 1.35·19-s + 3.56·31-s + 0.833·49-s + 0.764·61-s + 4·64-s − 2.70·76-s − 2.51·79-s + 1.13·109-s − 2·121-s + 7.13·124-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 − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$50625$$    =    $$3^{4} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$176.237$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 50625,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$4.491012790$$ $$L(\frac12)$$ $$\approx$$ $$4.491012790$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
5 $$1$$
good2$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
7$C_2^2$ $$1 - 286 T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$1 + 506 T^{2} + p^{6} T^{4}$$
17$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
19$C_2$ $$( 1 + 56 T + p^{3} T^{2} )^{2}$$
23$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
29$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
31$C_2$ $$( 1 - 308 T + p^{3} T^{2} )^{2}$$
37$C_2^2$ $$1 - 89206 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$1 + 111386 T^{2} + p^{6} T^{4}$$
47$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
53$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
59$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 - 182 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 + 172874 T^{2} + p^{6} T^{4}$$
71$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 638066 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 + 884 T + p^{3} T^{2} )^{2}$$
83$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
89$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
97$C_2^2$ $$1 - 56446 T^{2} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$