# Properties

 Label 2-2e2-4.3-c8-0-2 Degree $2$ Conductor $4$ Sign $-0.218 + 0.975i$ Analytic cond. $1.62951$ Root an. cond. $1.27652$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−10 − 12.4i)2-s − 99.9i·3-s + (−56 + 249. i)4-s + 610·5-s + (−1.24e3 + 999. i)6-s − 1.39e3i·7-s + (3.68e3 − 1.79e3i)8-s − 3.42e3·9-s + (−6.10e3 − 7.61e3i)10-s + 1.84e4i·11-s + (2.49e4 + 5.59e3i)12-s − 5.47e3·13-s + (−1.74e4 + 1.39e4i)14-s − 6.09e4i·15-s + (−5.92e4 − 2.79e4i)16-s + 7.30e4·17-s + ⋯
 L(s)  = 1 + (−0.625 − 0.780i)2-s − 1.23i·3-s + (−0.218 + 0.975i)4-s + 0.976·5-s + (−0.962 + 0.770i)6-s − 0.582i·7-s + (0.898 − 0.439i)8-s − 0.521·9-s + (−0.609 − 0.761i)10-s + 1.26i·11-s + (1.20 + 0.269i)12-s − 0.191·13-s + (−0.454 + 0.364i)14-s − 1.20i·15-s + (−0.904 − 0.426i)16-s + 0.875·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4$$    =    $$2^{2}$$ Sign: $-0.218 + 0.975i$ Analytic conductor: $$1.62951$$ Root analytic conductor: $$1.27652$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{4} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4,\ (\ :4),\ -0.218 + 0.975i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$0.608015 - 0.759411i$$ $$L(\frac12)$$ $$\approx$$ $$0.608015 - 0.759411i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (10 + 12.4i)T$$
good3 $$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}$$