# Properties

 Label 4-3234e2-1.1-c1e2-0-6 Degree $4$ Conductor $10458756$ Sign $1$ Analytic cond. $666.859$ Root an. cond. $5.08169$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 4·8-s + 3·9-s − 4·10-s − 2·11-s − 6·12-s + 4·15-s + 5·16-s − 6·17-s + 6·18-s − 2·19-s − 6·20-s − 4·22-s + 4·23-s − 8·24-s − 2·25-s − 4·27-s + 8·30-s − 10·31-s + 6·32-s + 4·33-s − 12·34-s + 9·36-s + ⋯
 L(s)  = 1 + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s − 2/5·25-s − 0.769·27-s + 1.46·30-s − 1.79·31-s + 1.06·32-s + 0.696·33-s − 2.05·34-s + 3/2·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10458756$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$666.859$$ Root analytic conductor: $$5.08169$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3234} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 10458756,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_1$ $$( 1 - T )^{2}$$
3$C_1$ $$( 1 + T )^{2}$$
7 $$1$$
11$C_1$ $$( 1 + T )^{2}$$
good5$D_{4}$ $$1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 + p T^{2} )^{2}$$
17$D_{4}$ $$1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
19$C_4$ $$1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$D_{4}$ $$1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
61$C_2$ $$( 1 + p T^{2} )^{2}$$
67$D_{4}$ $$1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
71$C_4$ $$1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$