# Properties

 Degree $4$ Conductor $16384$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 4·5-s + 8·7-s + 6·9-s + 92·11-s − 100·13-s − 16·15-s + 92·17-s + 4·19-s + 32·21-s − 8·23-s − 46·25-s + 92·27-s − 84·29-s + 384·31-s + 368·33-s − 32·35-s + 172·37-s − 400·39-s − 300·41-s + 300·43-s − 24·45-s + 16·47-s − 446·49-s + 368·51-s + 12·53-s − 368·55-s + ⋯
 L(s)  = 1 + 0.769·3-s − 0.357·5-s + 0.431·7-s + 2/9·9-s + 2.52·11-s − 2.13·13-s − 0.275·15-s + 1.31·17-s + 0.0482·19-s + 0.332·21-s − 0.0725·23-s − 0.367·25-s + 0.655·27-s − 0.537·29-s + 2.22·31-s + 1.94·33-s − 0.154·35-s + 0.764·37-s − 1.64·39-s − 1.14·41-s + 1.06·43-s − 0.0795·45-s + 0.0496·47-s − 1.30·49-s + 1.01·51-s + 0.0311·53-s − 0.902·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$16384$$    =    $$2^{14}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{128} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 16384,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.08923$$ $$L(\frac12)$$ $$\approx$$ $$3.08923$$ $$L(\frac{5}{2})$$ 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)$
bad2 $$1$$
good3$D_{4}$ $$1 - 4 T + 10 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 + 4 T + 62 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 92 T + 430 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 100 T + 5166 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 92 T + 5030 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 4 T + 11370 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 8 T + 8798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 84 T + 45742 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 172 T + 99294 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 300 T + 157270 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 300 T + 180314 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 16 T + 114782 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 12 T + 288382 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 644 T + 462170 T^{2} + 644 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 292 T + 240078 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 172 T + 278250 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 408 T + 672766 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 948 T + 1360138 T^{2} - 948 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 572 T + 845846 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$