# Properties

 Label 4-21e4-1.1-c5e2-0-5 Degree $4$ Conductor $194481$ Sign $1$ Analytic cond. $5002.62$ Root an. cond. $8.41006$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s − 9·4-s − 72·5-s + 15·8-s − 216·10-s − 480·11-s + 1.29e3·13-s − 529·16-s − 936·17-s − 216·19-s + 648·20-s − 1.44e3·22-s + 504·23-s − 2.36e3·25-s + 3.88e3·26-s − 6.37e3·29-s + 9.93e3·31-s − 5.79e3·32-s − 2.80e3·34-s + 1.11e4·37-s − 648·38-s − 1.08e3·40-s − 2.09e4·41-s − 6.26e3·43-s + 4.32e3·44-s + 1.51e3·46-s − 7.92e3·47-s + ⋯
 L(s)  = 1 + 0.530·2-s − 0.281·4-s − 1.28·5-s + 0.0828·8-s − 0.683·10-s − 1.19·11-s + 2.12·13-s − 0.516·16-s − 0.785·17-s − 0.137·19-s + 0.362·20-s − 0.634·22-s + 0.198·23-s − 0.755·25-s + 1.12·26-s − 1.40·29-s + 1.85·31-s − 1.00·32-s − 0.416·34-s + 1.33·37-s − 0.0727·38-s − 0.106·40-s − 1.94·41-s − 0.516·43-s + 0.336·44-s + 0.105·46-s − 0.522·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$194481$$    =    $$3^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$5002.62$$ Root analytic conductor: $$8.41006$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 194481,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 $$1$$
7 $$1$$
good2$D_{4}$ $$1 - 3 T + 9 p T^{2} - 3 p^{5} T^{3} + p^{10} T^{4}$$
5$C_2$ $$( 1 + 36 T + p^{5} T^{2} )^{2}$$
11$D_{4}$ $$1 + 480 T + 376614 T^{2} + 480 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 1296 T + 912362 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 936 T + 807586 T^{2} + 936 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 216 T + 961814 T^{2} + 216 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 504 T + 12784878 T^{2} - 504 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 6372 T + 31409694 T^{2} + 6372 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 9936 T + 65931134 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 11124 T + 115818446 T^{2} - 11124 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 20952 T + 339207826 T^{2} + 20952 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 6264 T + 25906310 T^{2} + 6264 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 7920 T - 101923298 T^{2} + 7920 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 + 2220 T + 769983534 T^{2} + 2220 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 504 p T + 1614887590 T^{2} + 504 p^{6} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 17280 T + 707051402 T^{2} + 17280 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 20680 T + 11658642 p T^{2} + 20680 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 92280 T + 5423120334 T^{2} - 92280 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 56592 T + 4777720274 T^{2} - 56592 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 56096 T + 4914766302 T^{2} + 56096 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 71352 T + 4792627990 T^{2} - 71352 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 123192 T + 14311603186 T^{2} + 123192 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 35856 T - 1009376254 T^{2} - 35856 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$