# Properties

 Label 2-2e10-1.1-c1-0-5 Degree $2$ Conductor $1024$ Sign $1$ Analytic cond. $8.17668$ Root an. cond. $2.85948$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.61·3-s + 3.41·5-s − 1.53·7-s + 3.82·9-s + 4.77·11-s + 0.585·13-s − 8.92·15-s − 2.82·17-s + 0.448·19-s + 4·21-s − 5.86·23-s + 6.65·25-s − 2.16·27-s + 4.58·29-s + 7.39·31-s − 12.4·33-s − 5.22·35-s − 5.07·37-s − 1.53·39-s − 4·41-s + 2.61·43-s + 13.0·45-s + 7.39·47-s − 4.65·49-s + 7.39·51-s + 7.41·53-s + 16.3·55-s + ⋯
 L(s)  = 1 − 1.50·3-s + 1.52·5-s − 0.578·7-s + 1.27·9-s + 1.44·11-s + 0.162·13-s − 2.30·15-s − 0.685·17-s + 0.102·19-s + 0.872·21-s − 1.22·23-s + 1.33·25-s − 0.416·27-s + 0.851·29-s + 1.32·31-s − 2.17·33-s − 0.883·35-s − 0.833·37-s − 0.245·39-s − 0.624·41-s + 0.398·43-s + 1.94·45-s + 1.07·47-s − 0.665·49-s + 1.03·51-s + 1.01·53-s + 2.19·55-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$1024$$    =    $$2^{10}$$ Sign: $1$ Analytic conductor: $$8.17668$$ Root analytic conductor: $$2.85948$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1024} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1024,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.261448799$$ $$L(\frac12)$$ $$\approx$$ $$1.261448799$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 2.61T + 3T^{2}$$
5 $$1 - 3.41T + 5T^{2}$$
7 $$1 + 1.53T + 7T^{2}$$
11 $$1 - 4.77T + 11T^{2}$$
13 $$1 - 0.585T + 13T^{2}$$
17 $$1 + 2.82T + 17T^{2}$$
19 $$1 - 0.448T + 19T^{2}$$
23 $$1 + 5.86T + 23T^{2}$$
29 $$1 - 4.58T + 29T^{2}$$
31 $$1 - 7.39T + 31T^{2}$$
37 $$1 + 5.07T + 37T^{2}$$
41 $$1 + 4T + 41T^{2}$$
43 $$1 - 2.61T + 43T^{2}$$
47 $$1 - 7.39T + 47T^{2}$$
53 $$1 - 7.41T + 53T^{2}$$
59 $$1 + 2.61T + 59T^{2}$$
61 $$1 - 13.0T + 61T^{2}$$
67 $$1 - 10.0T + 67T^{2}$$
71 $$1 - 11.9T + 71T^{2}$$
73 $$1 - 10.4T + 73T^{2}$$
79 $$1 - 6.12T + 79T^{2}$$
83 $$1 + 3.50T + 83T^{2}$$
89 $$1 + 0.828T + 89T^{2}$$
97 $$1 - 10.8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$