# Properties

 Label 2-2e6-4.3-c8-0-3 Degree $2$ Conductor $64$ Sign $1$ Analytic cond. $26.0722$ Root an. cond. $5.10609$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 99.9i·3-s − 610·5-s + 1.39e3i·7-s − 3.42e3·9-s + 1.84e4i·11-s + 5.47e3·13-s + 6.09e4i·15-s + 7.30e4·17-s + 1.94e4i·19-s + 1.39e5·21-s − 2.37e5i·23-s − 1.85e4·25-s − 3.13e5i·27-s + 1.28e5·29-s − 6.79e4i·31-s + ⋯
 L(s)  = 1 − 1.23i·3-s − 0.976·5-s + 0.582i·7-s − 0.521·9-s + 1.26i·11-s + 0.191·13-s + 1.20i·15-s + 0.875·17-s + 0.149i·19-s + 0.718·21-s − 0.847i·23-s − 0.0474·25-s − 0.589i·27-s + 0.181·29-s − 0.0735i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$64$$    =    $$2^{6}$$ Sign: $1$ Analytic conductor: $$26.0722$$ Root analytic conductor: $$5.10609$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{64} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 64,\ (\ :4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.47545$$ $$L(\frac12)$$ $$\approx$$ $$1.47545$$ $$L(5)$$ 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 + 99.9iT - 6.56e3T^{2}$$
5 $$1 + 610T + 3.90e5T^{2}$$
7 $$1 - 1.39e3iT - 5.76e6T^{2}$$
11 $$1 - 1.84e4iT - 2.14e8T^{2}$$
13 $$1 - 5.47e3T + 8.15e8T^{2}$$
17 $$1 - 7.30e4T + 6.97e9T^{2}$$
19 $$1 - 1.94e4iT - 1.69e10T^{2}$$
23 $$1 + 2.37e5iT - 7.83e10T^{2}$$
29 $$1 - 1.28e5T + 5.00e11T^{2}$$
31 $$1 + 6.79e4iT - 8.52e11T^{2}$$
37 $$1 - 3.47e6T + 3.51e12T^{2}$$
41 $$1 - 2.14e6T + 7.98e12T^{2}$$
43 $$1 - 5.92e6iT - 1.16e13T^{2}$$
47 $$1 - 7.62e6iT - 2.38e13T^{2}$$
53 $$1 + 8.24e5T + 6.22e13T^{2}$$
59 $$1 - 3.72e6iT - 1.46e14T^{2}$$
61 $$1 - 1.47e7T + 1.91e14T^{2}$$
67 $$1 + 1.52e7iT - 4.06e14T^{2}$$
71 $$1 + 1.19e6iT - 6.45e14T^{2}$$
73 $$1 + 5.72e6T + 8.06e14T^{2}$$
79 $$1 - 3.59e7iT - 1.51e15T^{2}$$
83 $$1 - 5.19e7iT - 2.25e15T^{2}$$
89 $$1 + 8.33e7T + 3.93e15T^{2}$$
97 $$1 - 1.20e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$