# Properties

 Label 4-35e2-1.1-c5e2-0-0 Degree $4$ Conductor $1225$ Sign $1$ Analytic cond. $31.5106$ Root an. cond. $2.36926$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 3·3-s − 47·4-s − 50·5-s + 3·6-s − 98·7-s − 63·8-s − 333·9-s − 50·10-s − 601·11-s − 141·12-s − 577·13-s − 98·14-s − 150·15-s + 1.20e3·16-s + 41·17-s − 333·18-s + 630·19-s + 2.35e3·20-s − 294·21-s − 601·22-s − 442·23-s − 189·24-s + 1.87e3·25-s − 577·26-s − 1.29e3·27-s + 4.60e3·28-s + ⋯
 L(s)  = 1 + 0.176·2-s + 0.192·3-s − 1.46·4-s − 0.894·5-s + 0.0340·6-s − 0.755·7-s − 0.348·8-s − 1.37·9-s − 0.158·10-s − 1.49·11-s − 0.282·12-s − 0.946·13-s − 0.133·14-s − 0.172·15-s + 1.17·16-s + 0.0344·17-s − 0.242·18-s + 0.400·19-s + 1.31·20-s − 0.145·21-s − 0.264·22-s − 0.174·23-s − 0.0669·24-s + 3/5·25-s − 0.167·26-s − 0.342·27-s + 1.11·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$31.5106$$ Root analytic conductor: $$2.36926$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 1225,\ (\ :5/2, 5/2),\ 1)$$

## 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)$
bad5$C_1$ $$( 1 + p^{2} T )^{2}$$
7$C_1$ $$( 1 + p^{2} T )^{2}$$
good2$D_{4}$ $$1 - T + 3 p^{4} T^{2} - p^{5} T^{3} + p^{10} T^{4}$$
3$D_{4}$ $$1 - p T + 38 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 + 601 T + 259506 T^{2} + 601 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 577 T + 780172 T^{2} + 577 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 41 T + 1816368 T^{2} - 41 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 630 T + 4931238 T^{2} - 630 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 442 T - 299538 T^{2} + 442 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 - 5885 T + 35168948 T^{2} - 5885 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 396 T + 43423646 T^{2} + 396 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 + 8904 T + 132709718 T^{2} + 8904 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 1774 T - 4379094 T^{2} - 1774 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 27122 T + 464103742 T^{2} + 27122 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 21289 T + 531685238 T^{2} + 21289 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 + 55582 T + 1605381282 T^{2} + 55582 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 59600 T + 1881188438 T^{2} - 59600 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 51846 T + 1832119946 T^{2} + 51846 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 45344 T + 2875187158 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 80744 T + 4781205326 T^{2} - 80744 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 13532 T + 3906362902 T^{2} + 13532 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 51795 T + 1771153398 T^{2} + 51795 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 109828 T + 10643414822 T^{2} - 109828 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 37650 T + 10990453658 T^{2} + 37650 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 96339 T + 16335863448 T^{2} + 96339 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$