# Properties

 Label 4-1078e2-1.1-c1e2-0-26 Degree $4$ Conductor $1162084$ Sign $1$ Analytic cond. $74.0954$ Root an. cond. $2.93391$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 2·3-s + 2·5-s − 2·6-s + 8-s + 3·9-s − 2·10-s − 11-s + 4·13-s + 4·15-s − 16-s − 3·18-s + 2·19-s + 22-s + 2·24-s + 5·25-s − 4·26-s + 10·27-s + 12·29-s − 4·30-s − 4·31-s − 2·33-s − 2·37-s − 2·38-s + 8·39-s + 2·40-s + 16·41-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.15·3-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 9-s − 0.632·10-s − 0.301·11-s + 1.10·13-s + 1.03·15-s − 1/4·16-s − 0.707·18-s + 0.458·19-s + 0.213·22-s + 0.408·24-s + 25-s − 0.784·26-s + 1.92·27-s + 2.22·29-s − 0.730·30-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.324·38-s + 1.28·39-s + 0.316·40-s + 2.49·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1162084$$    =    $$2^{2} \cdot 7^{4} \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$74.0954$$ Root analytic conductor: $$2.93391$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1162084,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.333937939$$ $$L(\frac12)$$ $$\approx$$ $$3.333937939$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + T + T^{2}$$
7 $$1$$
11$C_2$ $$1 + T + T^{2}$$
good3$C_2^2$ $$1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
17$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
23$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
37$C_2^2$ $$1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
73$C_2^2$ $$1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 + 18 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$