# Properties

 Degree $4$ Conductor $3969$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 8·4-s − 3·5-s − 7·7-s − 45·8-s + 9·10-s − 15·11-s − 128·13-s + 21·14-s + 135·16-s + 84·17-s + 16·19-s − 24·20-s + 45·22-s − 84·23-s + 125·25-s + 384·26-s − 56·28-s + 594·29-s + 253·31-s − 360·32-s − 252·34-s + 21·35-s + 316·37-s − 48·38-s + 135·40-s − 720·41-s + ⋯
 L(s)  = 1 − 1.06·2-s + 4-s − 0.268·5-s − 0.377·7-s − 1.98·8-s + 0.284·10-s − 0.411·11-s − 2.73·13-s + 0.400·14-s + 2.10·16-s + 1.19·17-s + 0.193·19-s − 0.268·20-s + 0.436·22-s − 0.761·23-s + 25-s + 2.89·26-s − 0.377·28-s + 3.80·29-s + 1.46·31-s − 1.98·32-s − 1.27·34-s + 0.101·35-s + 1.40·37-s − 0.204·38-s + 0.533·40-s − 2.74·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$3969$$    =    $$3^{4} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{63} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 3969,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.611094$$ $$L(\frac12)$$ $$\approx$$ $$0.611094$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
7$C_2$ $$1 + p T + p^{3} T^{2}$$
good2$C_2^2$ $$1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}$$
5$C_2^2$ $$1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4}$$
13$C_2$ $$( 1 + 64 T + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2$ $$( 1 - 297 T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 - 253 T + 34218 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4}$$
37$C_2^2$ $$1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2$ $$( 1 + 360 T + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 - 26 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 363 T - 17108 T^{2} - 363 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2^2$ $$1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2^2$ $$1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4}$$
67$C_2^2$ $$1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 - 342 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4}$$
79$C_2^2$ $$1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4}$$
83$C_2$ $$( 1 + 477 T + p^{3} T^{2} )^{2}$$
89$C_2^2$ $$1 - 906 T + 115867 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2$ $$( 1 - 503 T + p^{3} T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$