# Properties

 Label 2-2e6-8.5-c1-0-0 Degree $2$ Conductor $64$ Sign $0.707 - 0.707i$ Analytic cond. $0.511042$ Root an. cond. $0.714872$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2i·3-s − 9-s − 6i·11-s − 6·17-s + 2i·19-s + 5·25-s + 4i·27-s + 12·33-s − 6·41-s + 10i·43-s − 7·49-s − 12i·51-s − 4·57-s − 6i·59-s − 14i·67-s + ⋯
 L(s)  = 1 + 1.15i·3-s − 0.333·9-s − 1.80i·11-s − 1.45·17-s + 0.458i·19-s + 25-s + 0.769i·27-s + 2.08·33-s − 0.937·41-s + 1.52i·43-s − 49-s − 1.68i·51-s − 0.529·57-s − 0.781i·59-s − 1.71i·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$64$$    =    $$2^{6}$$ Sign: $0.707 - 0.707i$ Analytic conductor: $$0.511042$$ Root analytic conductor: $$0.714872$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{64} (33, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 64,\ (\ :1/2),\ 0.707 - 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.822784 + 0.340808i$$ $$L(\frac12)$$ $$\approx$$ $$0.822784 + 0.340808i$$ $$L(\frac{3}{2})$$ 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 - 2iT - 3T^{2}$$
5 $$1 - 5T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 + 6iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 - 2iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 - 10iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 6iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 14iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 2T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 18iT - 83T^{2}$$
89 $$1 - 18T + 89T^{2}$$
97 $$1 - 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$