# Properties

 Label 2-252-63.20-c3-0-9 Degree $2$ Conductor $252$ Sign $0.850 + 0.525i$ 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 + (−4.09 − 3.19i)3-s + (3.53 + 6.12i)5-s + (10.8 − 15.0i)7-s + (6.56 + 26.1i)9-s + (7.40 + 4.27i)11-s + (−45.3 + 26.1i)13-s + (5.08 − 36.3i)15-s + 38.9·17-s + 66.4i·19-s + (−92.3 + 26.9i)21-s + (173. − 100. i)23-s + (37.5 − 64.9i)25-s + (56.7 − 128. i)27-s + (52.9 + 30.5i)29-s + (116. − 67.2i)31-s + ⋯
 L(s)  = 1 + (−0.788 − 0.615i)3-s + (0.316 + 0.547i)5-s + (0.584 − 0.811i)7-s + (0.243 + 0.969i)9-s + (0.202 + 0.117i)11-s + (−0.967 + 0.558i)13-s + (0.0875 − 0.626i)15-s + 0.555·17-s + 0.801i·19-s + (−0.960 + 0.279i)21-s + (1.57 − 0.910i)23-s + (0.300 − 0.519i)25-s + (0.404 − 0.914i)27-s + (0.339 + 0.195i)29-s + (0.674 − 0.389i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.850 + 0.525i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (209, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ 0.850 + 0.525i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.536638900$$ $$L(\frac12)$$ $$\approx$$ $$1.536638900$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (4.09 + 3.19i)T$$
7 $$1 + (-10.8 + 15.0i)T$$
good5 $$1 + (-3.53 - 6.12i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-7.40 - 4.27i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + (45.3 - 26.1i)T + (1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 38.9T + 4.91e3T^{2}$$
19 $$1 - 66.4iT - 6.85e3T^{2}$$
23 $$1 + (-173. + 100. i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-52.9 - 30.5i)T + (1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-116. + 67.2i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 - 298.T + 5.06e4T^{2}$$
41 $$1 + (221. + 383. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-26.1 + 45.2i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (-137. + 238. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + 136. iT - 1.48e5T^{2}$$
59 $$1 + (-191. - 331. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-261. - 151. i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-318. - 552. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 228. iT - 3.57e5T^{2}$$
73 $$1 - 1.24e3iT - 3.89e5T^{2}$$
79 $$1 + (100. - 174. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-323. + 560. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 826.T + 7.04e5T^{2}$$
97 $$1 + (17.0 + 9.85i)T + (4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$