# Properties

 Label 4-10005-1.1-c1e2-0-0 Degree $4$ Conductor $10005$ Sign $-1$ Analytic cond. $0.637927$ Root an. cond. $0.893702$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 2·4-s − 5-s + 7-s + 3·8-s − 2·9-s + 10-s − 3·11-s − 14-s + 16-s − 3·17-s + 2·18-s − 6·19-s + 2·20-s + 3·22-s + 3·23-s − 2·25-s + 3·27-s − 2·28-s − 17·31-s − 2·32-s + 3·34-s − 35-s + 4·36-s + 6·38-s − 3·40-s − 7·41-s + ⋯
 L(s)  = 1 − 0.707·2-s − 4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.639·22-s + 0.625·23-s − 2/5·25-s + 0.577·27-s − 0.377·28-s − 3.05·31-s − 0.353·32-s + 0.514·34-s − 0.169·35-s + 2/3·36-s + 0.973·38-s − 0.474·40-s − 1.09·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10005$$    =    $$3 \cdot 5 \cdot 23 \cdot 29$$ Sign: $-1$ Analytic conductor: $$0.637927$$ Root analytic conductor: $$0.893702$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{10005} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 10005,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad3$C_1$$\times$$C_2$ $$( 1 - T )( 1 + T + p T^{2} )$$
5$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 2 T + p T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 2 T + p T^{2} )$$
29$C_1$$\times$$C_2$ $$( 1 + T )( 1 - T + p T^{2} )$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$C_2^2$ $$1 - 3 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$C_2^2$ $$1 - 42 T^{2} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$D_{4}$ $$1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
61$C_4$ $$1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$D_{4}$ $$1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$