# Properties

 Label 4-525e2-1.1-c3e2-0-13 Degree $4$ Conductor $275625$ Sign $1$ Analytic cond. $959.512$ Root an. cond. $5.56560$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 6·3-s − 5·4-s − 18·6-s + 14·7-s + 33·8-s + 27·9-s − 31·11-s − 30·12-s − 39·13-s − 42·14-s − 21·16-s + 79·17-s − 81·18-s − 56·19-s + 84·21-s + 93·22-s − 254·23-s + 198·24-s + 117·26-s + 108·27-s − 70·28-s − 62·29-s − 135·31-s − 87·32-s − 186·33-s − 237·34-s + ⋯
 L(s)  = 1 − 1.06·2-s + 1.15·3-s − 5/8·4-s − 1.22·6-s + 0.755·7-s + 1.45·8-s + 9-s − 0.849·11-s − 0.721·12-s − 0.832·13-s − 0.801·14-s − 0.328·16-s + 1.12·17-s − 1.06·18-s − 0.676·19-s + 0.872·21-s + 0.901·22-s − 2.30·23-s + 1.68·24-s + 0.882·26-s + 0.769·27-s − 0.472·28-s − 0.397·29-s − 0.782·31-s − 0.480·32-s − 0.981·33-s − 1.19·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$275625$$    =    $$3^{2} \cdot 5^{4} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$959.512$$ Root analytic conductor: $$5.56560$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{525} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 275625,\ (\ :3/2, 3/2),\ 1)$$

## 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)$
bad3$C_1$ $$( 1 - p T )^{2}$$
5 $$1$$
7$C_1$ $$( 1 - p T )^{2}$$
good2$C_4$ $$1 + 3 T + 7 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 31 T + 2796 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 3 p T + 3546 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 79 T + 8288 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 254 T + 38015 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 62 T + 19751 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 113 T + 102964 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 235 T + 143042 T^{2} - 235 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 804 T + 306321 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 152 T + 180510 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 149 T + 269640 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 441 T + 273734 T^{2} + 441 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 1157 T + 871890 T^{2} + 1157 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 619 T + 576906 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 268 T + 763078 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 1211 T + 720720 T^{2} + 1211 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 466 T + 306170 T^{2} - 466 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 172 T - 273626 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$