# Properties

 Label 4-475e2-1.1-c2e2-0-1 Degree $4$ Conductor $225625$ Sign $1$ Analytic cond. $167.516$ Root an. cond. $3.59761$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·4-s − 18·9-s + 6·11-s + 48·16-s + 38·19-s + 144·36-s − 48·44-s + 73·49-s + 206·61-s − 256·64-s − 304·76-s + 243·81-s − 108·99-s − 204·101-s − 215·121-s + 127-s + 131-s + 137-s + 139-s − 864·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s − 684·171-s + ⋯
 L(s)  = 1 − 2·4-s − 2·9-s + 6/11·11-s + 3·16-s + 2·19-s + 4·36-s − 1.09·44-s + 1.48·49-s + 3.37·61-s − 4·64-s − 4·76-s + 3·81-s − 1.09·99-s − 2.01·101-s − 1.77·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s − 4·171-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$225625$$    =    $$5^{4} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$167.516$$ Root analytic conductor: $$3.59761$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{475} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 225625,\ (\ :1, 1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.9439256316$$ $$L(\frac12)$$ $$\approx$$ $$0.9439256316$$ $$L(2)$$ 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)$
bad5 $$1$$
19$C_1$ $$( 1 - p T )^{2}$$
good2$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
3$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
7$C_2^2$ $$1 - 73 T^{2} + p^{4} T^{4}$$
11$C_2$ $$( 1 - 3 T + p^{2} T^{2} )^{2}$$
13$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
17$C_2^2$ $$1 - 353 T^{2} + p^{4} T^{4}$$
23$C_2^2$ $$1 - 158 T^{2} + p^{4} T^{4}$$
29$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
37$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
43$C_2^2$ $$1 + 3527 T^{2} + p^{4} T^{4}$$
47$C_2^2$ $$1 + 1207 T^{2} + p^{4} T^{4}$$
53$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
59$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
61$C_2$ $$( 1 - 103 T + p^{2} T^{2} )^{2}$$
67$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
73$C_2^2$ $$1 - 10033 T^{2} + p^{4} T^{4}$$
79$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
83$C_2^2$ $$1 - 5678 T^{2} + p^{4} T^{4}$$
89$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
97$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$