# Properties

 Label 4-1575e2-1.1-c3e2-0-9 Degree $4$ Conductor $2480625$ Sign $1$ Analytic cond. $8635.61$ Root an. cond. $9.63991$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·4-s + 14·7-s − 164·13-s − 55·16-s − 40·19-s + 42·28-s + 312·31-s − 372·37-s − 328·43-s + 147·49-s − 492·52-s + 1.58e3·61-s − 357·64-s + 88·67-s − 252·73-s − 120·76-s − 1.42e3·79-s − 2.29e3·91-s − 1.59e3·97-s + 1.83e3·103-s − 684·109-s − 770·112-s − 762·121-s + 936·124-s + 127-s + 131-s − 560·133-s + ⋯
 L(s)  = 1 + 3/8·4-s + 0.755·7-s − 3.49·13-s − 0.859·16-s − 0.482·19-s + 0.283·28-s + 1.80·31-s − 1.65·37-s − 1.16·43-s + 3/7·49-s − 1.31·52-s + 3.31·61-s − 0.697·64-s + 0.160·67-s − 0.404·73-s − 0.181·76-s − 2.02·79-s − 2.64·91-s − 1.67·97-s + 1.75·103-s − 0.601·109-s − 0.649·112-s − 0.572·121-s + 0.677·124-s + 0.000698·127-s + 0.000666·131-s − 0.365·133-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2480625$$    =    $$3^{4} \cdot 5^{4} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$8635.61$$ Root analytic conductor: $$9.63991$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1575} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 2480625,\ (\ :3/2, 3/2),\ 1)$$

## 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 $$1$$
5 $$1$$
7$C_1$ $$( 1 - p T )^{2}$$
good2$C_2^2$ $$1 - 3 T^{2} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 762 T^{2} + p^{6} T^{4}$$
13$C_2$ $$( 1 + 82 T + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 + 3670 T^{2} + p^{6} T^{4}$$
19$C_2$ $$( 1 + 20 T + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 + 7234 T^{2} + p^{6} T^{4}$$
29$C_2^2$ $$1 - 10806 T^{2} + p^{6} T^{4}$$
31$C_2$ $$( 1 - 156 T + p^{3} T^{2} )^{2}$$
37$C_2$ $$( 1 + 186 T + p^{3} T^{2} )^{2}$$
41$C_2^2$ $$1 + 110406 T^{2} + p^{6} T^{4}$$
43$C_2$ $$( 1 + 164 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 - 13970 T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 + 273130 T^{2} + p^{6} T^{4}$$
59$C_2^2$ $$1 + 386134 T^{2} + p^{6} T^{4}$$
61$C_2$ $$( 1 - 790 T + p^{3} T^{2} )^{2}$$
67$C_2$ $$( 1 - 44 T + p^{3} T^{2} )^{2}$$
71$C_2^2$ $$1 + 518146 T^{2} + p^{6} T^{4}$$
73$C_2$ $$( 1 + 126 T + p^{3} T^{2} )^{2}$$
79$C_2$ $$( 1 + 712 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 - 1001450 T^{2} + p^{6} T^{4}$$
89$C_2^2$ $$1 - 709626 T^{2} + p^{6} T^{4}$$
97$C_2$ $$( 1 + 798 T + p^{3} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$