# Properties

 Label 4-36992-1.1-c1e2-0-15 Degree $4$ Conductor $36992$ Sign $-1$ Analytic cond. $2.35864$ Root an. cond. $1.23926$ 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 − 8·13-s + 16-s + 2·17-s + 2·18-s − 4·19-s − 6·25-s + 8·26-s − 32-s − 2·34-s − 2·36-s + 4·38-s − 4·43-s − 2·49-s + 6·50-s − 8·52-s + 8·53-s − 20·59-s + 64-s − 4·67-s + 2·68-s + 2·72-s − 4·76-s − 5·81-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s − 6/5·25-s + 1.56·26-s − 0.176·32-s − 0.342·34-s − 1/3·36-s + 0.648·38-s − 0.609·43-s − 2/7·49-s + 0.848·50-s − 1.10·52-s + 1.09·53-s − 2.60·59-s + 1/8·64-s − 0.488·67-s + 0.242·68-s + 0.235·72-s − 0.458·76-s − 5/9·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$36992$$    =    $$2^{7} \cdot 17^{2}$$ Sign: $-1$ Analytic conductor: $$2.35864$$ Root analytic conductor: $$1.23926$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 36992,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_1$ $$1 + T$$
17$C_2$ $$1 - 2 T + p T^{2}$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - p T^{2} )^{2}$$
37$C_2^2$ $$1 - 58 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 - 34 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$