# Properties

 Label 4-901404-1.1-c1e2-0-4 Degree $4$ Conductor $901404$ Sign $-1$ Analytic cond. $57.4743$ Root an. cond. $2.75339$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s − 7-s − 3·9-s − 6·13-s + 16-s + 8·25-s + 28-s − 6·31-s + 3·36-s + 8·37-s + 8·43-s + 49-s + 6·52-s − 12·61-s + 3·63-s − 64-s + 4·67-s + 17·73-s − 2·79-s + 9·81-s + 6·91-s − 8·100-s − 18·103-s − 28·109-s − 112-s + 18·117-s + 4·121-s + ⋯
 L(s)  = 1 − 1/2·4-s − 0.377·7-s − 9-s − 1.66·13-s + 1/4·16-s + 8/5·25-s + 0.188·28-s − 1.07·31-s + 1/2·36-s + 1.31·37-s + 1.21·43-s + 1/7·49-s + 0.832·52-s − 1.53·61-s + 0.377·63-s − 1/8·64-s + 0.488·67-s + 1.98·73-s − 0.225·79-s + 81-s + 0.628·91-s − 4/5·100-s − 1.77·103-s − 2.68·109-s − 0.0944·112-s + 1.66·117-s + 4/11·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$901404$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73$$ Sign: $-1$ Analytic conductor: $$57.4743$$ Root analytic conductor: $$2.75339$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{901404} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 901404,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad2$C_2$ $$1 + T^{2}$$
3$C_2$ $$1 + p T^{2}$$
7$C_1$ $$1 + T$$
73$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 16 T + p T^{2} )$$
good5$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 62 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2^2$ $$1 + 70 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$