Properties

 Label 4-156800-1.1-c1e2-0-36 Degree $4$ Conductor $156800$ Sign $-1$ Analytic cond. $9.99770$ Root an. cond. $1.77817$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

Origins

Dirichlet series

 L(s)  = 1 − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s − 6·17-s + 2·18-s + 25-s + 2·28-s − 8·31-s − 32-s + 6·34-s − 2·36-s − 6·41-s − 6·47-s + 3·49-s − 50-s − 2·56-s + 8·62-s − 4·63-s + 64-s − 6·68-s + 2·72-s + 4·73-s − 20·79-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 1/5·25-s + 0.377·28-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 1/3·36-s − 0.937·41-s − 0.875·47-s + 3/7·49-s − 0.141·50-s − 0.267·56-s + 1.01·62-s − 0.503·63-s + 1/8·64-s − 0.727·68-s + 0.235·72-s + 0.468·73-s − 2.25·79-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$156800$$    =    $$2^{7} \cdot 5^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$9.99770$$ Root analytic conductor: $$1.77817$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{156800} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 156800,\ (\ :1/2, 1/2),\ -1)$$

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$C_1$ $$1 + T$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7$C_1$ $$( 1 - T )^{2}$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
11$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
37$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 62 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 94 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 98 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
83$C_2^2$ $$1 + 94 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$