# Properties

 Label 4-450e2-1.1-c5e2-0-21 Degree $4$ Conductor $202500$ Sign $1$ Analytic cond. $5208.90$ Root an. cond. $8.49545$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·2-s + 48·4-s + 100·7-s + 256·8-s − 540·11-s − 890·13-s + 800·14-s + 1.28e3·16-s − 492·17-s + 592·19-s − 4.32e3·22-s − 3.66e3·23-s − 7.12e3·26-s + 4.80e3·28-s − 5.70e3·29-s − 5.70e3·31-s + 6.14e3·32-s − 3.93e3·34-s − 1.13e4·37-s + 4.73e3·38-s − 1.54e4·41-s − 6.32e3·43-s − 2.59e4·44-s − 2.92e4·46-s + 7.80e3·47-s + 1.06e4·49-s − 4.27e4·52-s + ⋯
 L(s)  = 1 + 1.41·2-s + 3/2·4-s + 0.771·7-s + 1.41·8-s − 1.34·11-s − 1.46·13-s + 1.09·14-s + 5/4·16-s − 0.412·17-s + 0.376·19-s − 1.90·22-s − 1.44·23-s − 2.06·26-s + 1.15·28-s − 1.25·29-s − 1.06·31-s + 1.06·32-s − 0.583·34-s − 1.35·37-s + 0.532·38-s − 1.43·41-s − 0.521·43-s − 2.01·44-s − 2.04·46-s + 0.515·47-s + 0.631·49-s − 2.19·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$202500$$    =    $$2^{2} \cdot 3^{4} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$5208.90$$ Root analytic conductor: $$8.49545$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{450} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 202500,\ (\ :5/2, 5/2),\ 1)$$

## 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)$
bad2$C_1$ $$( 1 - p^{2} T )^{2}$$
3 $$1$$
5 $$1$$
good7$D_{4}$ $$1 - 100 T - 615 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 + 540 T + 248086 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 890 T + 793695 T^{2} + 890 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 492 T - 772670 T^{2} + 492 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 592 T + 4121589 T^{2} - 592 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 3660 T + 12548686 T^{2} + 3660 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 5700 T + 48997882 T^{2} + 5700 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 5708 T + 64485393 T^{2} + 5708 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 + 11300 T + 149454510 T^{2} + 11300 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 15420 T + 258100402 T^{2} + 15420 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 6320 T + 242261037 T^{2} + 6320 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 7800 T + 106610014 T^{2} - 7800 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 27828 T + 850018282 T^{2} - 27828 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 50520 T + 1968600982 T^{2} + 50520 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 29126 T + 1603768671 T^{2} + 29126 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 97400 T + 5071609653 T^{2} + 97400 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 6180 T + 3034897198 T^{2} + 6180 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 32900 T + 3887848086 T^{2} + 32900 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 7912 T + 1409684334 T^{2} - 7912 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 163464 T + 14190911110 T^{2} - 163464 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 164640 T + 13798144114 T^{2} + 164640 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 52430 T + 12995461155 T^{2} + 52430 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$