# Properties

 Label 2-126-7.5-c2-0-4 Degree $2$ Conductor $126$ Sign $-0.605 + 0.795i$ Analytic cond. $3.43325$ Root an. cond. $1.85290$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.24 − 2.44i)5-s + (3.5 − 6.06i)7-s − 2.82·8-s + (−6 + 3.46i)10-s + (−8.48 − 14.6i)11-s + 1.73i·13-s + (−4.94 − 8.57i)14-s + (−2.00 + 3.46i)16-s + (4.24 − 2.44i)17-s + (25.5 + 14.7i)19-s + 9.79i·20-s − 24·22-s + (−4.24 + 7.34i)23-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.848 − 0.489i)5-s + (0.5 − 0.866i)7-s − 0.353·8-s + (−0.600 + 0.346i)10-s + (−0.771 − 1.33i)11-s + 0.133i·13-s + (−0.353 − 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.249 − 0.144i)17-s + (1.34 + 0.774i)19-s + 0.489i·20-s − 1.09·22-s + (−0.184 + 0.319i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $-0.605 + 0.795i$ Analytic conductor: $$3.43325$$ Root analytic conductor: $$1.85290$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{126} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 126,\ (\ :1),\ -0.605 + 0.795i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.581636 - 1.17329i$$ $$L(\frac12)$$ $$\approx$$ $$0.581636 - 1.17329i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.707 + 1.22i)T$$
3 $$1$$
7 $$1 + (-3.5 + 6.06i)T$$
good5 $$1 + (4.24 + 2.44i)T + (12.5 + 21.6i)T^{2}$$
11 $$1 + (8.48 + 14.6i)T + (-60.5 + 104. i)T^{2}$$
13 $$1 - 1.73iT - 169T^{2}$$
17 $$1 + (-4.24 + 2.44i)T + (144.5 - 250. i)T^{2}$$
19 $$1 + (-25.5 - 14.7i)T + (180.5 + 312. i)T^{2}$$
23 $$1 + (4.24 - 7.34i)T + (-264.5 - 458. i)T^{2}$$
29 $$1 - 33.9T + 841T^{2}$$
31 $$1 + (10.5 - 6.06i)T + (480.5 - 832. i)T^{2}$$
37 $$1 + (-23.5 + 40.7i)T + (-684.5 - 1.18e3i)T^{2}$$
41 $$1 - 68.5iT - 1.68e3T^{2}$$
43 $$1 - 31T + 1.84e3T^{2}$$
47 $$1 + (-72.1 - 41.6i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 + (38.1 + 66.1i)T + (-1.40e3 + 2.43e3i)T^{2}$$
59 $$1 + (-72.1 + 41.6i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (72 + 41.5i)T + (1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 + 59.3T + 5.04e3T^{2}$$
73 $$1 + (70.5 - 40.7i)T + (2.66e3 - 4.61e3i)T^{2}$$
79 $$1 + (20.5 - 35.5i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + 4.89iT - 6.88e3T^{2}$$
89 $$1 + (50.9 + 29.3i)T + (3.96e3 + 6.85e3i)T^{2}$$
97 $$1 + 41.5iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$