# Properties

 Label 2-252-7.4-c3-0-7 Degree $2$ Conductor $252$ Sign $-0.269 + 0.963i$ 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 + (−0.723 − 1.25i)5-s + (−12.3 + 13.7i)7-s + (23.0 − 39.9i)11-s − 32.2·13-s + (38.8 − 67.3i)17-s + (−6.33 − 10.9i)19-s + (−50.4 − 87.4i)23-s + (61.4 − 106. i)25-s − 213.·29-s + (−21.0 + 36.4i)31-s + (26.1 + 5.56i)35-s + (−155. − 268. i)37-s − 44.0·41-s + 381.·43-s + (−179. − 310. i)47-s + ⋯
 L(s)  = 1 + (−0.0646 − 0.112i)5-s + (−0.669 + 0.743i)7-s + (0.632 − 1.09i)11-s − 0.687·13-s + (0.554 − 0.961i)17-s + (−0.0764 − 0.132i)19-s + (−0.457 − 0.792i)23-s + (0.491 − 0.851i)25-s − 1.36·29-s + (−0.121 + 0.211i)31-s + (0.126 + 0.0268i)35-s + (−0.688 − 1.19i)37-s − 0.167·41-s + 1.35·43-s + (−0.555 − 0.962i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$252$$    =    $$2^{2} \cdot 3^{2} \cdot 7$$ Sign: $-0.269 + 0.963i$ Analytic conductor: $$14.8684$$ Root analytic conductor: $$3.85596$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{252} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 252,\ (\ :3/2),\ -0.269 + 0.963i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.025539713$$ $$L(\frac12)$$ $$\approx$$ $$1.025539713$$ $$L(\frac{5}{2})$$ 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 + (12.3 - 13.7i)T$$
good5 $$1 + (0.723 + 1.25i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-23.0 + 39.9i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + 32.2T + 2.19e3T^{2}$$
17 $$1 + (-38.8 + 67.3i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (6.33 + 10.9i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (50.4 + 87.4i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 213.T + 2.43e4T^{2}$$
31 $$1 + (21.0 - 36.4i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (155. + 268. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 44.0T + 6.89e4T^{2}$$
43 $$1 - 381.T + 7.95e4T^{2}$$
47 $$1 + (179. + 310. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-92.4 + 160. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-227. + 393. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (5.92 + 10.2i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (295. - 511. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 494.T + 3.57e5T^{2}$$
73 $$1 + (487. - 844. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (149. + 259. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 1.40e3T + 5.71e5T^{2}$$
89 $$1 + (347. + 602. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 481.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$