# Properties

 Degree 2 Conductor $2^{10}$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 8.43·3-s − 12.2·5-s + 1.63·7-s + 44.1·9-s + 25.7·11-s + 13.2·13-s + 103.·15-s − 53.6·17-s − 100.·19-s − 13.8·21-s + 25.1·23-s + 25.6·25-s − 144.·27-s − 256.·29-s + 132.·31-s − 217.·33-s − 20.1·35-s − 247.·37-s − 111.·39-s + 198.·41-s − 404.·43-s − 542.·45-s − 78.3·47-s − 340.·49-s + 452.·51-s − 743.·53-s − 315.·55-s + ⋯
 L(s)  = 1 − 1.62·3-s − 1.09·5-s + 0.0885·7-s + 1.63·9-s + 0.705·11-s + 0.282·13-s + 1.78·15-s − 0.764·17-s − 1.21·19-s − 0.143·21-s + 0.227·23-s + 0.205·25-s − 1.03·27-s − 1.63·29-s + 0.768·31-s − 1.14·33-s − 0.0971·35-s − 1.09·37-s − 0.457·39-s + 0.756·41-s − 1.43·43-s − 1.79·45-s − 0.243·47-s − 0.992·49-s + 1.24·51-s − 1.92·53-s − 0.774·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1024$$    =    $$2^{10}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{1024} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 1024,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$0.3825353315$$ $$L(\frac12)$$ $$\approx$$ $$0.3825353315$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 8.43T + 27T^{2}$$
5 $$1 + 12.2T + 125T^{2}$$
7 $$1 - 1.63T + 343T^{2}$$
11 $$1 - 25.7T + 1.33e3T^{2}$$
13 $$1 - 13.2T + 2.19e3T^{2}$$
17 $$1 + 53.6T + 4.91e3T^{2}$$
19 $$1 + 100.T + 6.85e3T^{2}$$
23 $$1 - 25.1T + 1.21e4T^{2}$$
29 $$1 + 256.T + 2.43e4T^{2}$$
31 $$1 - 132.T + 2.97e4T^{2}$$
37 $$1 + 247.T + 5.06e4T^{2}$$
41 $$1 - 198.T + 6.89e4T^{2}$$
43 $$1 + 404.T + 7.95e4T^{2}$$
47 $$1 + 78.3T + 1.03e5T^{2}$$
53 $$1 + 743.T + 1.48e5T^{2}$$
59 $$1 + 65.8T + 2.05e5T^{2}$$
61 $$1 - 273.T + 2.26e5T^{2}$$
67 $$1 - 399.T + 3.00e5T^{2}$$
71 $$1 - 727.T + 3.57e5T^{2}$$
73 $$1 - 106.T + 3.89e5T^{2}$$
79 $$1 - 58.9T + 4.93e5T^{2}$$
83 $$1 - 580.T + 5.71e5T^{2}$$
89 $$1 + 768.T + 7.04e5T^{2}$$
97 $$1 + 809.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}