# Properties

 Label 2-252-63.41-c3-0-17 Degree $2$ Conductor $252$ Sign $-0.00752 + 0.999i$ 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 + (−3.66 − 3.68i)3-s + (8.29 − 14.3i)5-s + (18.2 + 3.26i)7-s + (−0.127 + 26.9i)9-s + (46.2 − 26.7i)11-s + (11.0 + 6.37i)13-s + (−83.3 + 22.1i)15-s + 96.7·17-s + 54.6i·19-s + (−54.7 − 79.1i)21-s + (−55.6 − 32.1i)23-s + (−75.2 − 130. i)25-s + (99.9 − 98.5i)27-s + (−112. + 65.0i)29-s + (−190. − 110. i)31-s + ⋯
 L(s)  = 1 + (−0.705 − 0.708i)3-s + (0.742 − 1.28i)5-s + (0.984 + 0.176i)7-s + (−0.00470 + 0.999i)9-s + (1.26 − 0.732i)11-s + (0.235 + 0.136i)13-s + (−1.43 + 0.380i)15-s + 1.38·17-s + 0.660i·19-s + (−0.569 − 0.822i)21-s + (−0.504 − 0.291i)23-s + (−0.601 − 1.04i)25-s + (0.712 − 0.702i)27-s + (−0.721 + 0.416i)29-s + (−1.10 − 0.637i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00752 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00752 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $-0.00752 + 0.999i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (41, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ -0.00752 + 0.999i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.948444006$$ $$L(\frac12)$$ $$\approx$$ $$1.948444006$$ $$L(\frac{5}{2})$$ 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 + (3.66 + 3.68i)T$$
7 $$1 + (-18.2 - 3.26i)T$$
good5 $$1 + (-8.29 + 14.3i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-46.2 + 26.7i)T + (665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-11.0 - 6.37i)T + (1.09e3 + 1.90e3i)T^{2}$$
17 $$1 - 96.7T + 4.91e3T^{2}$$
19 $$1 - 54.6iT - 6.85e3T^{2}$$
23 $$1 + (55.6 + 32.1i)T + (6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (112. - 65.0i)T + (1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (190. + 110. i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 279.T + 5.06e4T^{2}$$
41 $$1 + (-185. + 320. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (153. + 266. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (163. + 283. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 - 451. iT - 1.48e5T^{2}$$
59 $$1 + (258. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (234. - 135. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (370. - 642. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 914. iT - 3.57e5T^{2}$$
73 $$1 + 337. iT - 3.89e5T^{2}$$
79 $$1 + (498. + 863. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (17.0 + 29.4i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 - 208.T + 7.04e5T^{2}$$
97 $$1 + (1.10e3 - 635. i)T + (4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$