# Properties

 Label 4-1148e2-1.1-c1e2-0-3 Degree $4$ Conductor $1317904$ Sign $-1$ Analytic cond. $84.0307$ Root an. cond. $3.02767$ 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 − 4·5-s − 3·8-s + 3·9-s − 4·10-s − 2·13-s − 16-s + 4·17-s + 3·18-s + 4·20-s + 3·25-s − 2·26-s − 2·29-s + 5·32-s + 4·34-s − 3·36-s + 4·37-s + 12·40-s − 12·41-s − 12·45-s − 49-s + 3·50-s + 2·52-s + 10·53-s − 2·58-s + 4·61-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 9-s − 1.26·10-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.894·20-s + 3/5·25-s − 0.392·26-s − 0.371·29-s + 0.883·32-s + 0.685·34-s − 1/2·36-s + 0.657·37-s + 1.89·40-s − 1.87·41-s − 1.78·45-s − 1/7·49-s + 0.424·50-s + 0.277·52-s + 1.37·53-s − 0.262·58-s + 0.512·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1317904$$    =    $$2^{4} \cdot 7^{2} \cdot 41^{2}$$ Sign: $-1$ Analytic conductor: $$84.0307$$ Root analytic conductor: $$3.02767$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1317904} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 1317904,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_2$ $$1 - T + p T^{2}$$
7$C_2$ $$1 + T^{2}$$
41$C_2$ $$1 + 12 T + p T^{2}$$
good3$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )$$
11$C_2^2$ $$1 + 5 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 + T + p T^{2} )^{2}$$
17$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
19$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$C_2^2$ $$1 - 27 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
43$C_2^2$ $$1 + 55 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 39 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
59$C_2^2$ $$1 + 115 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
67$C_2^2$ $$1 + 97 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 73 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2^2$ $$1 - 89 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 91 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$