# Properties

 Label 4-35e2-1.1-c3e2-0-1 Degree $4$ Conductor $1225$ Sign $1$ Analytic cond. $4.26450$ Root an. cond. $1.43703$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·2-s + 2·3-s + 34·4-s − 10·5-s + 16·6-s − 14·7-s + 96·8-s − 19·9-s − 80·10-s − 14·11-s + 68·12-s + 50·13-s − 112·14-s − 20·15-s + 196·16-s − 50·17-s − 152·18-s + 36·19-s − 340·20-s − 28·21-s − 112·22-s + 244·23-s + 192·24-s + 75·25-s + 400·26-s − 30·27-s − 476·28-s + ⋯
 L(s)  = 1 + 2.82·2-s + 0.384·3-s + 17/4·4-s − 0.894·5-s + 1.08·6-s − 0.755·7-s + 4.24·8-s − 0.703·9-s − 2.52·10-s − 0.383·11-s + 1.63·12-s + 1.06·13-s − 2.13·14-s − 0.344·15-s + 3.06·16-s − 0.713·17-s − 1.99·18-s + 0.434·19-s − 3.80·20-s − 0.290·21-s − 1.08·22-s + 2.21·23-s + 1.63·24-s + 3/5·25-s + 3.01·26-s − 0.213·27-s − 3.21·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$4.26450$$ Root analytic conductor: $$1.43703$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1225,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$5.223089146$$ $$L(\frac12)$$ $$\approx$$ $$5.223089146$$ $$L(\frac{5}{2})$$ 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$C_1$ $$( 1 + p T )^{2}$$
7$C_1$ $$( 1 + p T )^{2}$$
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$D_{4}$ $$1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$