# Properties

 Label 4-320e2-1.1-c2e2-0-7 Degree $4$ Conductor $102400$ Sign $1$ Analytic cond. $76.0273$ Root an. cond. $2.95285$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 4·7-s + 8·9-s + 16·11-s − 6·13-s + 14·17-s + 16·21-s − 4·23-s − 25·25-s + 36·27-s + 104·31-s + 64·33-s + 6·37-s − 24·39-s − 16·41-s + 84·43-s − 36·47-s + 8·49-s + 56·51-s − 106·53-s + 96·61-s + 32·63-s − 124·67-s − 16·69-s − 56·71-s − 94·73-s − 100·75-s + ⋯
 L(s)  = 1 + 4/3·3-s + 4/7·7-s + 8/9·9-s + 1.45·11-s − 0.461·13-s + 0.823·17-s + 0.761·21-s − 0.173·23-s − 25-s + 4/3·27-s + 3.35·31-s + 1.93·33-s + 6/37·37-s − 0.615·39-s − 0.390·41-s + 1.95·43-s − 0.765·47-s + 8/49·49-s + 1.09·51-s − 2·53-s + 1.57·61-s + 0.507·63-s − 1.85·67-s − 0.231·69-s − 0.788·71-s − 1.28·73-s − 4/3·75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$102400$$    =    $$2^{12} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$76.0273$$ Root analytic conductor: $$2.95285$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 102400,\ (\ :1, 1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$4.521283195$$ $$L(\frac12)$$ $$\approx$$ $$4.521283195$$ $$L(2)$$ 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$C_2$ $$1 + p^{2} T^{2}$$
good3$C_2^2$ $$1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4}$$
7$C_2^2$ $$1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4}$$
11$C_2$ $$( 1 - 8 T + p^{2} T^{2} )^{2}$$
13$C_2^2$ $$1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4}$$
17$C_2$ $$( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} )$$
19$C_2^2$ $$1 - 322 T^{2} + p^{4} T^{4}$$
23$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4}$$
29$C_2$ $$( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} )$$
31$C_2$ $$( 1 - 52 T + p^{2} T^{2} )^{2}$$
37$C_2^2$ $$1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4}$$
41$C_2$ $$( 1 + 8 T + p^{2} T^{2} )^{2}$$
43$C_2^2$ $$1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4}$$
47$C_2^2$ $$1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4}$$
53$C_1$$\times$$C_2$ $$( 1 + p T )^{2}( 1 + p^{2} T^{2} )$$
59$C_2^2$ $$1 - 6562 T^{2} + p^{4} T^{4}$$
61$C_2$ $$( 1 - 48 T + p^{2} T^{2} )^{2}$$
67$C_2^2$ $$1 + 124 T + 7688 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4}$$
71$C_2$ $$( 1 + 28 T + p^{2} T^{2} )^{2}$$
73$C_2^2$ $$1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4}$$
79$C_1$$\times$$C_1$ $$( 1 - p T )^{2}( 1 + p T )^{2}$$
83$C_2^2$ $$1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4}$$
89$C_2^2$ $$1 - 9442 T^{2} + p^{4} T^{4}$$
97$C_2^2$ $$1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$