# Properties

 Label 4-10e4-1.1-c1e2-0-1 Degree $4$ Conductor $10000$ Sign $1$ Analytic cond. $0.637608$ Root an. cond. $0.893590$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·11-s + 4·31-s − 16·41-s − 16·61-s + 4·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 + 1.20·11-s + 0.718·31-s − 2.49·41-s − 2.04·61-s + 0.474·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10000$$    =    $$2^{4} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$0.637608$$ Root analytic conductor: $$0.893590$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 10000,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.031407104$$ $$L(\frac12)$$ $$\approx$$ $$1.031407104$$ $$L(\frac{3}{2})$$ 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 $$1$$
5 $$1$$
good3$C_2^2$ $$1 + p^{2} T^{4}$$
7$C_2^2$ $$1 + p^{2} T^{4}$$
11$C_4$ $$1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + p^{2} T^{4}$$
17$C_2^2$ $$1 + p^{2} T^{4}$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2^2$ $$1 + p^{2} T^{4}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_4$ $$1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
37$C_2^2$ $$1 + p^{2} T^{4}$$
41$C_4$ $$1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 + p^{2} T^{4}$$
47$C_2^2$ $$1 + p^{2} T^{4}$$
53$C_2^2$ $$1 + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_4$ $$1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 + p^{2} T^{4}$$
71$C_4$ $$1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
73$C_2^2$ $$1 + p^{2} T^{4}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2^2$ $$1 + p^{2} T^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$C_2^2$ $$1 + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$