# 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 − 4.62·3-s + 17.8·5-s + 13.8·7-s − 5.59·9-s + 2.18·11-s − 46.3·13-s − 82.7·15-s − 18.6·17-s + 122.·19-s − 64.1·21-s + 134.·23-s + 194.·25-s + 150.·27-s − 84.5·29-s − 31.5·31-s − 10.1·33-s + 248.·35-s − 126.·37-s + 214.·39-s + 210.·41-s + 168.·43-s − 100.·45-s + 182.·47-s − 150.·49-s + 86.2·51-s − 37.0·53-s + 39.1·55-s + ⋯
 L(s)  = 1 − 0.890·3-s + 1.59·5-s + 0.749·7-s − 0.207·9-s + 0.0599·11-s − 0.989·13-s − 1.42·15-s − 0.266·17-s + 1.47·19-s − 0.667·21-s + 1.21·23-s + 1.55·25-s + 1.07·27-s − 0.541·29-s − 0.182·31-s − 0.0533·33-s + 1.19·35-s − 0.560·37-s + 0.880·39-s + 0.801·41-s + 0.598·43-s − 0.331·45-s + 0.567·47-s − 0.438·49-s + 0.236·51-s − 0.0958·53-s + 0.0959·55-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 = $$(2,\ 1024,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$2.207526728$$ $$L(\frac12)$$ $$\approx$$ $$2.207526728$$ $$L(\frac{5}{2})$$ 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 + 4.62T + 27T^{2}$$
5 $$1 - 17.8T + 125T^{2}$$
7 $$1 - 13.8T + 343T^{2}$$
11 $$1 - 2.18T + 1.33e3T^{2}$$
13 $$1 + 46.3T + 2.19e3T^{2}$$
17 $$1 + 18.6T + 4.91e3T^{2}$$
19 $$1 - 122.T + 6.85e3T^{2}$$
23 $$1 - 134.T + 1.21e4T^{2}$$
29 $$1 + 84.5T + 2.43e4T^{2}$$
31 $$1 + 31.5T + 2.97e4T^{2}$$
37 $$1 + 126.T + 5.06e4T^{2}$$
41 $$1 - 210.T + 6.89e4T^{2}$$
43 $$1 - 168.T + 7.95e4T^{2}$$
47 $$1 - 182.T + 1.03e5T^{2}$$
53 $$1 + 37.0T + 1.48e5T^{2}$$
59 $$1 + 624.T + 2.05e5T^{2}$$
61 $$1 + 246.T + 2.26e5T^{2}$$
67 $$1 - 129.T + 3.00e5T^{2}$$
71 $$1 - 348.T + 3.57e5T^{2}$$
73 $$1 - 299.T + 3.89e5T^{2}$$
79 $$1 - 943.T + 4.93e5T^{2}$$
83 $$1 + 443.T + 5.71e5T^{2}$$
89 $$1 - 1.41e3T + 7.04e5T^{2}$$
97 $$1 - 1.51e3T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}