Properties

 Label 2-84-7.3-c8-0-4 Degree $2$ Conductor $84$ Sign $0.214 - 0.976i$ 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 + (1.01e3 − 584. i)5-s + (−1.82e3 + 1.55e3i)7-s + (1.09e3 + 1.89e3i)9-s + (−1.30e4 + 2.26e4i)11-s + 2.85e4i·13-s + 5.46e4·15-s + (−1.80e4 − 1.04e4i)17-s + (4.29e4 − 2.48e4i)19-s + (−1.10e5 + 2.01e4i)21-s + (1.70e5 + 2.96e5i)23-s + (4.87e5 − 8.43e5i)25-s + 1.02e5i·27-s − 1.12e6·29-s + (1.47e6 + 8.51e5i)31-s + ⋯
 L(s)  = 1 + (0.5 + 0.288i)3-s + (1.61 − 0.934i)5-s + (−0.762 + 0.647i)7-s + (0.166 + 0.288i)9-s + (−0.893 + 1.54i)11-s + 1.00i·13-s + 1.07·15-s + (−0.215 − 0.124i)17-s + (0.329 − 0.190i)19-s + (−0.567 + 0.103i)21-s + (0.610 + 1.05i)23-s + (1.24 − 2.16i)25-s + 0.192i·27-s − 1.59·29-s + (1.59 + 0.922i)31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$84$$    =    $$2^{2} \cdot 3 \cdot 7$$ Sign: $0.214 - 0.976i$ Analytic conductor: $$34.2198$$ Root analytic conductor: $$5.84976$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{84} (73, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 84,\ (\ :4),\ 0.214 - 0.976i)$$

Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.99820 + 1.60702i$$ $$L(\frac12)$$ $$\approx$$ $$1.99820 + 1.60702i$$ $$L(5)$$ 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 + (1.82e3 - 1.55e3i)T$$
good5 $$1 + (-1.01e3 + 584. i)T + (1.95e5 - 3.38e5i)T^{2}$$
11 $$1 + (1.30e4 - 2.26e4i)T + (-1.07e8 - 1.85e8i)T^{2}$$
13 $$1 - 2.85e4iT - 8.15e8T^{2}$$
17 $$1 + (1.80e4 + 1.04e4i)T + (3.48e9 + 6.04e9i)T^{2}$$
19 $$1 + (-4.29e4 + 2.48e4i)T + (8.49e9 - 1.47e10i)T^{2}$$
23 $$1 + (-1.70e5 - 2.96e5i)T + (-3.91e10 + 6.78e10i)T^{2}$$
29 $$1 + 1.12e6T + 5.00e11T^{2}$$
31 $$1 + (-1.47e6 - 8.51e5i)T + (4.26e11 + 7.38e11i)T^{2}$$
37 $$1 + (-3.20e5 - 5.54e5i)T + (-1.75e12 + 3.04e12i)T^{2}$$
41 $$1 - 1.34e6iT - 7.98e12T^{2}$$
43 $$1 + 1.80e6T + 1.16e13T^{2}$$
47 $$1 + (-7.88e6 + 4.55e6i)T + (1.19e13 - 2.06e13i)T^{2}$$
53 $$1 + (1.82e6 - 3.16e6i)T + (-3.11e13 - 5.39e13i)T^{2}$$
59 $$1 + (-9.97e6 - 5.76e6i)T + (7.34e13 + 1.27e14i)T^{2}$$
61 $$1 + (8.77e6 - 5.06e6i)T + (9.58e13 - 1.66e14i)T^{2}$$
67 $$1 + (8.86e6 - 1.53e7i)T + (-2.03e14 - 3.51e14i)T^{2}$$
71 $$1 + 3.79e7T + 6.45e14T^{2}$$
73 $$1 + (-2.94e7 - 1.70e7i)T + (4.03e14 + 6.98e14i)T^{2}$$
79 $$1 + (9.66e6 + 1.67e7i)T + (-7.58e14 + 1.31e15i)T^{2}$$
83 $$1 - 9.60e5iT - 2.25e15T^{2}$$
89 $$1 + (-5.62e7 + 3.24e7i)T + (1.96e15 - 3.40e15i)T^{2}$$
97 $$1 + 2.66e7iT - 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$