# Properties

 Label 2-3200-8.5-c1-0-17 Degree $2$ Conductor $3200$ Sign $-i$ Analytic cond. $25.5521$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.317i·3-s + 2.89·9-s + 3.78i·11-s − 1.89·17-s + 5.97i·19-s − 1.87i·27-s + 1.20·33-s − 6.79·41-s + 8.48i·43-s − 7·49-s + 0.603i·51-s + 1.89·57-s + 14.1i·59-s − 16.3i·67-s − 15.6·73-s + ⋯
 L(s)  = 1 − 0.183i·3-s + 0.966·9-s + 1.14i·11-s − 0.460·17-s + 1.37i·19-s − 0.360i·27-s + 0.209·33-s − 1.06·41-s + 1.29i·43-s − 49-s + 0.0845i·51-s + 0.251·57-s + 1.84i·59-s − 1.99i·67-s − 1.83·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $-i$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1601, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ -i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.494910580$$ $$L(\frac12)$$ $$\approx$$ $$1.494910580$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + 0.317iT - 3T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 - 3.78iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 1.89T + 17T^{2}$$
19 $$1 - 5.97iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 6.79T + 41T^{2}$$
43 $$1 - 8.48iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 - 14.1iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 16.3iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 15.6T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 17.0iT - 83T^{2}$$
89 $$1 - 4.10T + 89T^{2}$$
97 $$1 - 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$