# Properties

 Label 4-932261-1.1-c1e2-0-0 Degree $4$ Conductor $932261$ Sign $-1$ Analytic cond. $59.4417$ Root an. cond. $2.77666$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $3$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3-s + 4-s − 3·5-s + 2·6-s − 2·7-s + 6·10-s − 5·11-s − 12-s − 7·13-s + 4·14-s + 3·15-s + 16-s − 3·17-s − 4·19-s − 3·20-s + 2·21-s + 10·22-s − 5·23-s + 4·25-s + 14·26-s − 2·27-s − 2·28-s − 8·29-s − 6·30-s − 2·31-s + 2·32-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.755·7-s + 1.89·10-s − 1.50·11-s − 0.288·12-s − 1.94·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.670·20-s + 0.436·21-s + 2.13·22-s − 1.04·23-s + 4/5·25-s + 2.74·26-s − 0.384·27-s − 0.377·28-s − 1.48·29-s − 1.09·30-s − 0.359·31-s + 0.353·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 932261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 932261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$932261$$    =    $$11 \cdot 84751$$ Sign: $-1$ Analytic conductor: $$59.4417$$ Root analytic conductor: $$2.77666$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{932261} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(4,\ 932261,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad11$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 4 T + p T^{2} )$$
84751$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 395 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + T + T^{2} + p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} )$$
31$D_{4}$ $$1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
47$D_{4}$ $$1 + 16 T + 129 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 22 T + 222 T^{2} + 22 p T^{3} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$D_{4}$ $$1 - 7 T + 118 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 4 T - 68 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
97$D_{4}$ $$1 + 8 T + 141 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$