# Properties

 Label 2-252-28.27-c3-0-35 Degree $2$ Conductor $252$ Sign $0.987 - 0.158i$ Analytic cond. $14.8684$ Root an. cond. $3.85596$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.69 − 0.856i)2-s + (6.53 − 4.61i)4-s + 10.3i·5-s + (16.6 + 8.15i)7-s + (13.6 − 18.0i)8-s + (8.83 + 27.7i)10-s − 18.0i·11-s + 49.0i·13-s + (51.8 + 7.73i)14-s + (21.3 − 60.3i)16-s + 46.6i·17-s + 48.8·19-s + (47.6 + 67.3i)20-s + (−15.4 − 48.6i)22-s + 33.7i·23-s + ⋯
 L(s)  = 1 + (0.952 − 0.302i)2-s + (0.816 − 0.577i)4-s + 0.921i·5-s + (0.897 + 0.440i)7-s + (0.603 − 0.797i)8-s + (0.279 + 0.878i)10-s − 0.494i·11-s + 1.04i·13-s + (0.989 + 0.147i)14-s + (0.332 − 0.942i)16-s + 0.666i·17-s + 0.589·19-s + (0.532 + 0.752i)20-s + (−0.149 − 0.471i)22-s + 0.306i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.987 - 0.158i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (55, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ 0.987 - 0.158i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.641729728$$ $$L(\frac12)$$ $$\approx$$ $$3.641729728$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2.69 + 0.856i)T$$
3 $$1$$
7 $$1 + (-16.6 - 8.15i)T$$
good5 $$1 - 10.3iT - 125T^{2}$$
11 $$1 + 18.0iT - 1.33e3T^{2}$$
13 $$1 - 49.0iT - 2.19e3T^{2}$$
17 $$1 - 46.6iT - 4.91e3T^{2}$$
19 $$1 - 48.8T + 6.85e3T^{2}$$
23 $$1 - 33.7iT - 1.21e4T^{2}$$
29 $$1 - 48.1T + 2.43e4T^{2}$$
31 $$1 - 152.T + 2.97e4T^{2}$$
37 $$1 - 37.6T + 5.06e4T^{2}$$
41 $$1 + 409. iT - 6.89e4T^{2}$$
43 $$1 + 470. iT - 7.95e4T^{2}$$
47 $$1 + 548.T + 1.03e5T^{2}$$
53 $$1 + 203.T + 1.48e5T^{2}$$
59 $$1 + 717.T + 2.05e5T^{2}$$
61 $$1 - 493. iT - 2.26e5T^{2}$$
67 $$1 - 240. iT - 3.00e5T^{2}$$
71 $$1 - 995. iT - 3.57e5T^{2}$$
73 $$1 + 790. iT - 3.89e5T^{2}$$
79 $$1 - 214. iT - 4.93e5T^{2}$$
83 $$1 + 885.T + 5.71e5T^{2}$$
89 $$1 - 67.3iT - 7.04e5T^{2}$$
97 $$1 + 934. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$