# Properties

 Label 2-12e3-9.2-c2-0-15 Degree $2$ Conductor $1728$ Sign $0.743 - 0.668i$ Analytic cond. $47.0845$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−8.20 + 4.73i)5-s + (1.05 − 1.83i)7-s + (13.7 + 7.91i)11-s + (−4.70 − 8.14i)13-s − 11.6i·17-s − 12.9·19-s + (5.27 − 3.04i)23-s + (32.4 − 56.1i)25-s + (−24.7 − 14.2i)29-s + (8.75 + 15.1i)31-s + 20.0i·35-s + 15.6·37-s + (−14.8 + 8.54i)41-s + (21.7 − 37.6i)43-s + (−20.6 − 11.9i)47-s + ⋯
 L(s)  = 1 + (−1.64 + 0.947i)5-s + (0.150 − 0.261i)7-s + (1.24 + 0.719i)11-s + (−0.361 − 0.626i)13-s − 0.682i·17-s − 0.682·19-s + (0.229 − 0.132i)23-s + (1.29 − 2.24i)25-s + (−0.854 − 0.493i)29-s + (0.282 + 0.489i)31-s + 0.572i·35-s + 0.422·37-s + (−0.361 + 0.208i)41-s + (0.505 − 0.874i)43-s + (−0.439 − 0.253i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $0.743 - 0.668i$ Analytic conductor: $$47.0845$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1601, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :1),\ 0.743 - 0.668i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.249473178$$ $$L(\frac12)$$ $$\approx$$ $$1.249473178$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + (8.20 - 4.73i)T + (12.5 - 21.6i)T^{2}$$
7 $$1 + (-1.05 + 1.83i)T + (-24.5 - 42.4i)T^{2}$$
11 $$1 + (-13.7 - 7.91i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (4.70 + 8.14i)T + (-84.5 + 146. i)T^{2}$$
17 $$1 + 11.6iT - 289T^{2}$$
19 $$1 + 12.9T + 361T^{2}$$
23 $$1 + (-5.27 + 3.04i)T + (264.5 - 458. i)T^{2}$$
29 $$1 + (24.7 + 14.2i)T + (420.5 + 728. i)T^{2}$$
31 $$1 + (-8.75 - 15.1i)T + (-480.5 + 832. i)T^{2}$$
37 $$1 - 15.6T + 1.36e3T^{2}$$
41 $$1 + (14.8 - 8.54i)T + (840.5 - 1.45e3i)T^{2}$$
43 $$1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2}$$
47 $$1 + (20.6 + 11.9i)T + (1.10e3 + 1.91e3i)T^{2}$$
53 $$1 - 14.1iT - 2.80e3T^{2}$$
59 $$1 + (-38.5 + 22.2i)T + (1.74e3 - 3.01e3i)T^{2}$$
61 $$1 + (-1.86 + 3.22i)T + (-1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (21.0 + 36.3i)T + (-2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 120. iT - 5.04e3T^{2}$$
73 $$1 - 5.48T + 5.32e3T^{2}$$
79 $$1 + (-60.5 + 104. i)T + (-3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (-46.5 - 26.8i)T + (3.44e3 + 5.96e3i)T^{2}$$
89 $$1 - 102. iT - 7.92e3T^{2}$$
97 $$1 + (58.9 - 102. i)T + (-4.70e3 - 8.14e3i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$