# Properties

 Label 4-35e4-1.1-c3e2-0-5 Degree $4$ Conductor $1500625$ Sign $1$ Analytic cond. $5224.01$ Root an. cond. $8.50160$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·2-s + 2·3-s + 34·4-s − 16·6-s − 96·8-s − 19·9-s − 14·11-s + 68·12-s + 50·13-s + 196·16-s − 50·17-s + 152·18-s − 36·19-s + 112·22-s − 244·23-s − 192·24-s − 400·26-s − 30·27-s − 26·29-s + 120·31-s − 352·32-s − 28·33-s + 400·34-s − 646·36-s − 564·37-s + 288·38-s + 100·39-s + ⋯
 L(s)  = 1 − 2.82·2-s + 0.384·3-s + 17/4·4-s − 1.08·6-s − 4.24·8-s − 0.703·9-s − 0.383·11-s + 1.63·12-s + 1.06·13-s + 3.06·16-s − 0.713·17-s + 1.99·18-s − 0.434·19-s + 1.08·22-s − 2.21·23-s − 1.63·24-s − 3.01·26-s − 0.213·27-s − 0.166·29-s + 0.695·31-s − 1.94·32-s − 0.147·33-s + 2.01·34-s − 2.99·36-s − 2.50·37-s + 1.22·38-s + 0.410·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1500625$$    =    $$5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$5224.01$$ Root analytic conductor: $$8.50160$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1225} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1500625,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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)$
bad5 $$1$$
7 $$1$$
good2$D_{4}$ $$1 + p^{3} T + 15 p T^{2} + p^{6} T^{3} + p^{6} T^{4}$$
3$D_{4}$ $$1 - 2 T + 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 50 T + 4987 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 36 T + 10170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 244 T + 29970 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 120 T - 1618 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 564 T + 173630 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 260 T + 166666 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 350 T + 203423 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 56 T + 265770 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2$ $$( 1 - 616 T + p^{3} T^{2} )^{2}$$
61$D_{4}$ $$1 + 336 T + 458858 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 152 T + 599110 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 + 952 T + p^{3} T^{2} )^{2}$$
73$D_{4}$ $$1 - 676 T + 655606 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$