Properties

 Label 2-126-7.4-c3-0-7 Degree $2$ Conductor $126$ Sign $0.902 + 0.431i$ Analytic cond. $7.43424$ Root an. cond. $2.72658$ Motivic weight $3$ Arithmetic yes Rational no 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 + (−7.91 − 13.7i)5-s + (18.3 − 2.59i)7-s − 7.99·8-s + (15.8 − 27.4i)10-s + (25.9 − 44.8i)11-s + 38.8·13-s + (22.8 + 29.1i)14-s + (−8 − 13.8i)16-s + (13.6 − 23.6i)17-s + (−38.2 − 66.2i)19-s + 63.3·20-s + 103.·22-s + (73.6 + 127. i)23-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.708 − 1.22i)5-s + (0.990 − 0.140i)7-s − 0.353·8-s + (0.500 − 0.867i)10-s + (0.710 − 1.23i)11-s + 0.828·13-s + (0.435 + 0.556i)14-s + (−0.125 − 0.216i)16-s + (0.195 − 0.337i)17-s + (−0.461 − 0.800i)19-s + 0.708·20-s + 1.00·22-s + (0.667 + 1.15i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $0.902 + 0.431i$ Analytic conductor: $$7.43424$$ Root analytic conductor: $$2.72658$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{126} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 126,\ (\ :3/2),\ 0.902 + 0.431i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.77492 - 0.402713i$$ $$L(\frac12)$$ $$\approx$$ $$1.77492 - 0.402713i$$ $$L(\frac{5}{2})$$ 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 + (-18.3 + 2.59i)T$$
good5 $$1 + (7.91 + 13.7i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-25.9 + 44.8i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 38.8T + 2.19e3T^{2}$$
17 $$1 + (-13.6 + 23.6i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (38.2 + 66.2i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-73.6 - 127. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 240.T + 2.43e4T^{2}$$
31 $$1 + (-148. + 256. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (-80.7 - 139. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 102.T + 6.89e4T^{2}$$
43 $$1 + 328.T + 7.95e4T^{2}$$
47 $$1 + (-33.9 - 58.8i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (33.2 - 57.5i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (230. - 400. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (92.6 + 160. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (272. - 472. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 130.T + 3.57e5T^{2}$$
73 $$1 + (90.6 - 157. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-204. - 354. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 347.T + 5.71e5T^{2}$$
89 $$1 + (-578. - 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 1.61e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$