# Properties

 Label 2-252-21.5-c3-0-4 Degree $2$ Conductor $252$ Sign $0.898 - 0.438i$ 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 + (−4.36 + 7.56i)5-s + (14.4 − 11.5i)7-s + (7.60 − 4.39i)11-s − 11.8i·13-s + (22.2 + 38.6i)17-s + (10.0 + 5.82i)19-s + (123. + 71.3i)23-s + (24.3 + 42.1i)25-s + 234. i·29-s + (252. − 145. i)31-s + (23.9 + 160. i)35-s + (44.4 − 76.9i)37-s + 145.·41-s + 144.·43-s + (−120. + 208. i)47-s + ⋯
 L(s)  = 1 + (−0.390 + 0.676i)5-s + (0.782 − 0.622i)7-s + (0.208 − 0.120i)11-s − 0.252i·13-s + (0.318 + 0.550i)17-s + (0.121 + 0.0703i)19-s + (1.11 + 0.646i)23-s + (0.194 + 0.337i)25-s + 1.49i·29-s + (1.46 − 0.845i)31-s + (0.115 + 0.772i)35-s + (0.197 − 0.342i)37-s + 0.555·41-s + 0.512·43-s + (−0.372 + 0.646i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $0.898 - 0.438i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (89, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ 0.898 - 0.438i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.867037217$$ $$L(\frac12)$$ $$\approx$$ $$1.867037217$$ $$L(\frac{5}{2})$$ 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$$
7 $$1 + (-14.4 + 11.5i)T$$
good5 $$1 + (4.36 - 7.56i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-7.60 + 4.39i)T + (665.5 - 1.15e3i)T^{2}$$
13 $$1 + 11.8iT - 2.19e3T^{2}$$
17 $$1 + (-22.2 - 38.6i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-10.0 - 5.82i)T + (3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-123. - 71.3i)T + (6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 234. iT - 2.43e4T^{2}$$
31 $$1 + (-252. + 145. i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (-44.4 + 76.9i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 145.T + 6.89e4T^{2}$$
43 $$1 - 144.T + 7.95e4T^{2}$$
47 $$1 + (120. - 208. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-3.54 - 6.13i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (149. + 86.4i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-243. - 421. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 653. iT - 3.57e5T^{2}$$
73 $$1 + (99.0 - 57.1i)T + (1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (147. - 255. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 877.T + 5.71e5T^{2}$$
89 $$1 + (-710. + 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 738. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$