# Properties

 Label 4-758912-1.1-c1e2-0-65 Degree $4$ Conductor $758912$ Sign $-1$ Analytic cond. $48.3888$ Root an. cond. $2.63746$ 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 + 8-s + 2·9-s − 2·11-s + 16-s + 2·18-s − 2·22-s − 6·25-s + 32-s + 2·36-s − 24·43-s − 2·44-s − 7·49-s − 6·50-s + 64-s − 24·67-s + 2·72-s − 5·81-s − 24·86-s − 2·88-s − 7·98-s − 4·99-s − 6·100-s + 24·107-s + 20·113-s + 3·121-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s − 0.603·11-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 6/5·25-s + 0.176·32-s + 1/3·36-s − 3.65·43-s − 0.301·44-s − 49-s − 0.848·50-s + 1/8·64-s − 2.93·67-s + 0.235·72-s − 5/9·81-s − 2.58·86-s − 0.213·88-s − 0.707·98-s − 0.402·99-s − 3/5·100-s + 2.32·107-s + 1.88·113-s + 3/11·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$758912$$    =    $$2^{7} \cdot 7^{2} \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$48.3888$$ Root analytic conductor: $$2.63746$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 758912,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_1$ $$1 - T$$
7$C_2$ $$1 + p T^{2}$$
11$C_1$ $$( 1 + T )^{2}$$
good3$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - p T^{2} )^{2}$$
19$C_2^2$ $$1 - 34 T^{2} + p^{2} T^{4}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
53$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
59$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
73$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2^2$ $$1 + 158 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - p T^{2} )^{2}$$
97$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$