# Properties

 Label 4-68e2-1.1-c0e2-0-0 Degree $4$ Conductor $4624$ Sign $1$ Analytic cond. $0.00115168$ Root an. cond. $0.184218$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4-s − 2·5-s + 16-s − 2·17-s + 2·20-s + 2·25-s + 2·29-s − 2·37-s + 2·41-s − 2·61-s − 64-s + 2·68-s + 2·73-s − 2·80-s − 81-s + 4·85-s − 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯
 L(s)  = 1 − 4-s − 2·5-s + 16-s − 2·17-s + 2·20-s + 2·25-s + 2·29-s − 2·37-s + 2·41-s − 2·61-s − 64-s + 2·68-s + 2·73-s − 2·80-s − 81-s + 4·85-s − 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$4624$$    =    $$2^{4} \cdot 17^{2}$$ Sign: $1$ Analytic conductor: $$0.00115168$$ Root analytic conductor: $$0.184218$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{68} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 4624,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1773206590$$ $$L(\frac12)$$ $$\approx$$ $$0.1773206590$$ $$L(1)$$ 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$C_2$ $$1 + T^{2}$$
17$C_1$ $$( 1 + T )^{2}$$
good3$C_2^2$ $$1 + T^{4}$$
5$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
7$C_2^2$ $$1 + T^{4}$$
11$C_2^2$ $$1 + T^{4}$$
13$C_2$ $$( 1 + T^{2} )^{2}$$
19$C_2$ $$( 1 + T^{2} )^{2}$$
23$C_2^2$ $$1 + T^{4}$$
29$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
31$C_2^2$ $$1 + T^{4}$$
37$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
43$C_2$ $$( 1 + T^{2} )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_2$ $$( 1 + T^{2} )^{2}$$
61$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_2^2$ $$1 + T^{4}$$
73$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
79$C_2^2$ $$1 + T^{4}$$
83$C_2$ $$( 1 + T^{2} )^{2}$$
89$C_2$ $$( 1 + T^{2} )^{2}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$