# Properties

 Label 4-825e2-1.1-c5e2-0-1 Degree $4$ Conductor $680625$ Sign $1$ Analytic cond. $17507.6$ Root an. cond. $11.5028$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 13·2-s − 18·3-s + 71·4-s + 234·6-s − 146·7-s − 65·8-s + 243·9-s − 242·11-s − 1.27e3·12-s + 130·13-s + 1.89e3·14-s − 1.72e3·16-s + 728·17-s − 3.15e3·18-s − 828·19-s + 2.62e3·21-s + 3.14e3·22-s + 238·23-s + 1.17e3·24-s − 1.69e3·26-s − 2.91e3·27-s − 1.03e4·28-s + 696·29-s − 1.04e4·31-s + 1.26e4·32-s + 4.35e3·33-s − 9.46e3·34-s + ⋯
 L(s)  = 1 − 2.29·2-s − 1.15·3-s + 2.21·4-s + 2.65·6-s − 1.12·7-s − 0.359·8-s + 9-s − 0.603·11-s − 2.56·12-s + 0.213·13-s + 2.58·14-s − 1.68·16-s + 0.610·17-s − 2.29·18-s − 0.526·19-s + 1.30·21-s + 1.38·22-s + 0.0938·23-s + 0.414·24-s − 0.490·26-s − 0.769·27-s − 2.49·28-s + 0.153·29-s − 1.95·31-s + 2.17·32-s + 0.696·33-s − 1.40·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$680625$$    =    $$3^{2} \cdot 5^{4} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$17507.6$$ Root analytic conductor: $$11.5028$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 680625,\ (\ :5/2, 5/2),\ 1)$$

## 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$C_1$ $$( 1 + p^{2} T )^{2}$$
5 $$1$$
11$C_1$ $$( 1 + p^{2} T )^{2}$$
good2$D_{4}$ $$1 + 13 T + 49 p T^{2} + 13 p^{5} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 + 146 T + 7230 T^{2} + 146 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 10 p T + 701634 T^{2} - 10 p^{6} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 728 T + 1830542 T^{2} - 728 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 828 T - 865114 T^{2} + 828 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 238 T + 11988422 T^{2} - 238 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 - 24 p T + 22778374 T^{2} - 24 p^{6} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 10480 T + 63595902 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 1908 T + 74176718 T^{2} - 1908 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 36484 T + 564482438 T^{2} - 36484 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 9768 T + 237972854 T^{2} + 9768 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 43742 T + 935775830 T^{2} + 43742 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 12174 T + 457325722 T^{2} - 12174 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 2788 T + 1399448534 T^{2} + 2788 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 25302 T + 1815376826 T^{2} + 25302 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 40520 T + 1779236982 T^{2} - 40520 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 31386 T + 3698331094 T^{2} - 31386 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 46780 T + 4437463638 T^{2} - 46780 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 16850 T + 5148027246 T^{2} + 16850 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 79440 T + 5477266486 T^{2} + 79440 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 54204 T + 6019532470 T^{2} + 54204 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 241568 T + 30950947518 T^{2} - 241568 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$