Properties

 Label 4-442368-1.1-c1e2-0-29 Degree $4$ Conductor $442368$ Sign $-1$ Analytic cond. $28.2057$ Root an. cond. $2.30454$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

Origins

Dirichlet series

 L(s)  = 1 − 3-s + 9-s − 8·11-s + 6·25-s − 27-s + 8·33-s + 14·49-s + 8·59-s − 20·73-s − 6·75-s + 81-s + 8·83-s + 4·97-s − 8·99-s − 24·107-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 14·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1/3·9-s − 2.41·11-s + 6/5·25-s − 0.192·27-s + 1.39·33-s + 2·49-s + 1.04·59-s − 2.34·73-s − 0.692·75-s + 1/9·81-s + 0.878·83-s + 0.406·97-s − 0.804·99-s − 2.32·107-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.15·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$442368$$    =    $$2^{14} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$28.2057$$ Root analytic conductor: $$2.30454$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 442368,\ (\ :1/2, 1/2),\ -1)$$

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 $$1$$
3$C_1$ $$1 + T$$
good5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2^2$ $$1 - 102 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - 94 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$