# Properties

 Label 4-624e2-1.1-c3e2-0-5 Degree $4$ Conductor $389376$ Sign $1$ Analytic cond. $1355.50$ Root an. cond. $6.06771$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·3-s − 18·7-s − 90·11-s − 65·13-s + 117·17-s + 42·19-s + 54·21-s + 18·23-s + 223·25-s + 27·27-s + 99·29-s + 270·33-s + 195·37-s + 195·39-s − 63·41-s − 82·43-s − 127·49-s − 351·51-s − 522·53-s − 126·57-s + 1.36e3·59-s + 719·61-s + 1.21e3·67-s − 54·69-s + 810·71-s − 669·75-s + 1.62e3·77-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.971·7-s − 2.46·11-s − 1.38·13-s + 1.66·17-s + 0.507·19-s + 0.561·21-s + 0.163·23-s + 1.78·25-s + 0.192·27-s + 0.633·29-s + 1.42·33-s + 0.866·37-s + 0.800·39-s − 0.239·41-s − 0.290·43-s − 0.370·49-s − 0.963·51-s − 1.35·53-s − 0.292·57-s + 3.01·59-s + 1.50·61-s + 2.22·67-s − 0.0942·69-s + 1.35·71-s − 1.02·75-s + 2.39·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$389376$$    =    $$2^{8} \cdot 3^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$1355.50$$ Root analytic conductor: $$6.06771$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 389376,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.511831553$$ $$L(\frac12)$$ $$\approx$$ $$1.511831553$$ $$L(\frac{5}{2})$$ 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$C_2$ $$1 + p T + p^{2} T^{2}$$
13$C_2$ $$1 + 5 p T + p^{3} T^{2}$$
good5$C_2^2$ $$1 - 223 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 + 18 T + 451 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 90 T + 4031 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4}$$
17$C_2^2$ $$1 - 117 T + 8776 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 - 42 T + 7447 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 18 T - 11843 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2^2$ $$1 - 99 T - 14588 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4}$$
31$C_2^2$ $$1 - 21950 T^{2} + p^{6} T^{4}$$
37$C_2^2$ $$1 - 195 T + 63328 T^{2} - 195 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2^2$ $$1 + 63 T + 70244 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4}$$
43$C_2^2$ $$1 + 82 T - 72783 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4}$$
47$C_2^2$ $$1 - 202354 T^{2} + p^{6} T^{4}$$
53$C_2$ $$( 1 + 261 T + p^{3} T^{2} )^{2}$$
59$C_2^2$ $$1 - 1368 T + 829187 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2$ $$( 1 - 901 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} )$$
67$C_2^2$ $$1 - 1218 T + 795271 T^{2} - 1218 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2^2$ $$1 - 810 T + 576611 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4}$$
73$C_2^2$ $$1 - 309959 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 - 440 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 + 284726 T^{2} + p^{6} T^{4}$$
89$C_2^2$ $$1 - 2628 T + 3007097 T^{2} - 2628 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2^2$ $$1 + 2004 T + 2251345 T^{2} + 2004 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$