# Properties

 Degree $2$ Conductor $126$ Sign $-0.0165 + 0.999i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (3.5 + 6.06i)5-s + (−10 − 15.5i)7-s + 7.99·8-s + (7 − 12.1i)10-s + (17.5 − 30.3i)11-s + 66·13-s + (−17 + 32.9i)14-s + (−8 − 13.8i)16-s + (29.5 − 51.0i)17-s + (−68.5 − 118. i)19-s − 28·20-s − 70·22-s + (−3.5 − 6.06i)23-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.313 + 0.542i)5-s + (−0.539 − 0.841i)7-s + 0.353·8-s + (0.221 − 0.383i)10-s + (0.479 − 0.830i)11-s + 1.40·13-s + (−0.324 + 0.628i)14-s + (−0.125 − 0.216i)16-s + (0.420 − 0.728i)17-s + (−0.827 − 1.43i)19-s − 0.313·20-s − 0.678·22-s + (−0.0317 − 0.0549i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $-0.0165 + 0.999i$ Motivic weight: $$3$$ Character: $\chi_{126} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 126,\ (\ :3/2),\ -0.0165 + 0.999i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.895753 - 0.910745i$$ $$L(\frac12)$$ $$\approx$$ $$0.895753 - 0.910745i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1 + 1.73i)T$$
3 $$1$$
7 $$1 + (10 + 15.5i)T$$
good5 $$1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-17.5 + 30.3i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 66T + 2.19e3T^{2}$$
17 $$1 + (-29.5 + 51.0i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (68.5 + 118. i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 106T + 2.43e4T^{2}$$
31 $$1 + (37.5 - 64.9i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (5.5 + 9.52i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 498T + 6.89e4T^{2}$$
43 $$1 - 260T + 7.95e4T^{2}$$
47 $$1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (208.5 - 361. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (8.5 - 14.7i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (25.5 + 44.1i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (219.5 - 380. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 784T + 3.57e5T^{2}$$
73 $$1 + (147.5 - 255. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-247.5 - 428. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 932T + 5.71e5T^{2}$$
89 $$1 + (436.5 + 756. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 290T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$