# Properties

 Label 2-84-7.5-c8-0-3 Degree $2$ Conductor $84$ Sign $0.225 - 0.974i$ Analytic cond. $34.2198$ Root an. cond. $5.84976$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (40.5 − 23.3i)3-s + (632. + 365. i)5-s + (984. + 2.18e3i)7-s + (1.09e3 − 1.89e3i)9-s + (6.35e3 + 1.10e4i)11-s + 2.11e4i·13-s + 3.41e4·15-s + (−8.25e4 + 4.76e4i)17-s + (−1.33e5 − 7.69e4i)19-s + (9.10e4 + 6.56e4i)21-s + (−4.71e4 + 8.15e4i)23-s + (7.14e4 + 1.23e5i)25-s − 1.02e5i·27-s + 5.41e5·29-s + (−1.85e5 + 1.07e5i)31-s + ⋯
 L(s)  = 1 + (0.5 − 0.288i)3-s + (1.01 + 0.584i)5-s + (0.410 + 0.912i)7-s + (0.166 − 0.288i)9-s + (0.433 + 0.751i)11-s + 0.741i·13-s + 0.674·15-s + (−0.988 + 0.570i)17-s + (−1.02 − 0.590i)19-s + (0.468 + 0.337i)21-s + (−0.168 + 0.291i)23-s + (0.182 + 0.316i)25-s − 0.192i·27-s + 0.765·29-s + (−0.201 + 0.116i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$84$$    =    $$2^{2} \cdot 3 \cdot 7$$ Sign: $0.225 - 0.974i$ Analytic conductor: $$34.2198$$ Root analytic conductor: $$5.84976$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{84} (61, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 84,\ (\ :4),\ 0.225 - 0.974i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.16422 + 1.71975i$$ $$L(\frac12)$$ $$\approx$$ $$2.16422 + 1.71975i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-40.5 + 23.3i)T$$
7 $$1 + (-984. - 2.18e3i)T$$
good5 $$1 + (-632. - 365. i)T + (1.95e5 + 3.38e5i)T^{2}$$
11 $$1 + (-6.35e3 - 1.10e4i)T + (-1.07e8 + 1.85e8i)T^{2}$$
13 $$1 - 2.11e4iT - 8.15e8T^{2}$$
17 $$1 + (8.25e4 - 4.76e4i)T + (3.48e9 - 6.04e9i)T^{2}$$
19 $$1 + (1.33e5 + 7.69e4i)T + (8.49e9 + 1.47e10i)T^{2}$$
23 $$1 + (4.71e4 - 8.15e4i)T + (-3.91e10 - 6.78e10i)T^{2}$$
29 $$1 - 5.41e5T + 5.00e11T^{2}$$
31 $$1 + (1.85e5 - 1.07e5i)T + (4.26e11 - 7.38e11i)T^{2}$$
37 $$1 + (3.70e5 - 6.41e5i)T + (-1.75e12 - 3.04e12i)T^{2}$$
41 $$1 + 7.82e5iT - 7.98e12T^{2}$$
43 $$1 + 2.21e5T + 1.16e13T^{2}$$
47 $$1 + (-7.13e6 - 4.11e6i)T + (1.19e13 + 2.06e13i)T^{2}$$
53 $$1 + (-6.59e5 - 1.14e6i)T + (-3.11e13 + 5.39e13i)T^{2}$$
59 $$1 + (7.52e6 - 4.34e6i)T + (7.34e13 - 1.27e14i)T^{2}$$
61 $$1 + (-1.31e7 - 7.57e6i)T + (9.58e13 + 1.66e14i)T^{2}$$
67 $$1 + (-1.99e7 - 3.44e7i)T + (-2.03e14 + 3.51e14i)T^{2}$$
71 $$1 + 9.34e6T + 6.45e14T^{2}$$
73 $$1 + (-2.94e7 + 1.70e7i)T + (4.03e14 - 6.98e14i)T^{2}$$
79 $$1 + (-2.26e7 + 3.92e7i)T + (-7.58e14 - 1.31e15i)T^{2}$$
83 $$1 + 6.58e7iT - 2.25e15T^{2}$$
89 $$1 + (1.09e7 + 6.34e6i)T + (1.96e15 + 3.40e15i)T^{2}$$
97 $$1 + 6.15e6iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$