# Properties

 Label 2-252-3.2-c8-0-6 Degree $2$ Conductor $252$ Sign $-0.577 - 0.816i$ Analytic cond. $102.659$ Root an. cond. $10.1320$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 675. i·5-s + 907.·7-s + 1.89e4i·11-s + 6.23e3·13-s − 1.51e4i·17-s + 1.81e5·19-s + 4.21e5i·23-s − 6.60e4·25-s − 1.90e5i·29-s − 1.98e5·31-s + 6.13e5i·35-s + 7.00e5·37-s − 6.28e5i·41-s + 5.91e6·43-s + 6.30e6i·47-s + ⋯
 L(s)  = 1 + 1.08i·5-s + 0.377·7-s + 1.29i·11-s + 0.218·13-s − 0.181i·17-s + 1.39·19-s + 1.50i·23-s − 0.169·25-s − 0.268i·29-s − 0.214·31-s + 0.408i·35-s + 0.373·37-s − 0.222i·41-s + 1.73·43-s + 1.29i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $-0.577 - 0.816i$ Analytic conductor: $$102.659$$ Root analytic conductor: $$10.1320$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{252} (197, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :4),\ -0.577 - 0.816i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.180089361$$ $$L(\frac12)$$ $$\approx$$ $$2.180089361$$ $$L(5)$$ 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$$
7 $$1 - 907.T$$
good5 $$1 - 675. iT - 3.90e5T^{2}$$
11 $$1 - 1.89e4iT - 2.14e8T^{2}$$
13 $$1 - 6.23e3T + 8.15e8T^{2}$$
17 $$1 + 1.51e4iT - 6.97e9T^{2}$$
19 $$1 - 1.81e5T + 1.69e10T^{2}$$
23 $$1 - 4.21e5iT - 7.83e10T^{2}$$
29 $$1 + 1.90e5iT - 5.00e11T^{2}$$
31 $$1 + 1.98e5T + 8.52e11T^{2}$$
37 $$1 - 7.00e5T + 3.51e12T^{2}$$
41 $$1 + 6.28e5iT - 7.98e12T^{2}$$
43 $$1 - 5.91e6T + 1.16e13T^{2}$$
47 $$1 - 6.30e6iT - 2.38e13T^{2}$$
53 $$1 + 2.32e6iT - 6.22e13T^{2}$$
59 $$1 + 1.34e7iT - 1.46e14T^{2}$$
61 $$1 + 9.18e6T + 1.91e14T^{2}$$
67 $$1 - 6.29e6T + 4.06e14T^{2}$$
71 $$1 + 1.43e7iT - 6.45e14T^{2}$$
73 $$1 + 4.04e7T + 8.06e14T^{2}$$
79 $$1 + 3.00e7T + 1.51e15T^{2}$$
83 $$1 - 5.01e7iT - 2.25e15T^{2}$$
89 $$1 - 4.17e6iT - 3.93e15T^{2}$$
97 $$1 - 1.41e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$