# Properties

 Degree $2$ Conductor $22$ Sign $0.648 + 0.760i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.61 − 1.17i)2-s + (−1.33 − 4.09i)3-s + (1.23 − 3.80i)4-s + (6.52 + 4.73i)5-s + (−6.97 − 5.06i)6-s + (−8.05 + 24.8i)7-s + (−2.47 − 7.60i)8-s + (6.82 − 4.95i)9-s + 16.1·10-s + (−33.3 + 14.7i)11-s − 17.2·12-s + (2.64 − 1.91i)13-s + (16.1 + 49.6i)14-s + (10.7 − 33.0i)15-s + (−12.9 − 9.40i)16-s + (16.8 + 12.2i)17-s + ⋯
 L(s)  = 1 + (0.572 − 0.415i)2-s + (−0.256 − 0.788i)3-s + (0.154 − 0.475i)4-s + (0.583 + 0.423i)5-s + (−0.474 − 0.344i)6-s + (−0.435 + 1.33i)7-s + (−0.109 − 0.336i)8-s + (0.252 − 0.183i)9-s + 0.509·10-s + (−0.914 + 0.403i)11-s − 0.414·12-s + (0.0563 − 0.0409i)13-s + (0.307 + 0.946i)14-s + (0.184 − 0.568i)15-s + (−0.202 − 0.146i)16-s + (0.240 + 0.175i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$22$$    =    $$2 \cdot 11$$ Sign: $0.648 + 0.760i$ Motivic weight: $$3$$ Character: $\chi_{22} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 22,\ (\ :3/2),\ 0.648 + 0.760i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.25538 - 0.579360i$$ $$L(\frac12)$$ $$\approx$$ $$1.25538 - 0.579360i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.61 + 1.17i)T$$
11 $$1 + (33.3 - 14.7i)T$$
good3 $$1 + (1.33 + 4.09i)T + (-21.8 + 15.8i)T^{2}$$
5 $$1 + (-6.52 - 4.73i)T + (38.6 + 118. i)T^{2}$$
7 $$1 + (8.05 - 24.8i)T + (-277. - 201. i)T^{2}$$
13 $$1 + (-2.64 + 1.91i)T + (678. - 2.08e3i)T^{2}$$
17 $$1 + (-16.8 - 12.2i)T + (1.51e3 + 4.67e3i)T^{2}$$
19 $$1 + (38.9 + 119. i)T + (-5.54e3 + 4.03e3i)T^{2}$$
23 $$1 - 97.8T + 1.21e4T^{2}$$
29 $$1 + (81.5 - 250. i)T + (-1.97e4 - 1.43e4i)T^{2}$$
31 $$1 + (161. - 117. i)T + (9.20e3 - 2.83e4i)T^{2}$$
37 $$1 + (-112. + 347. i)T + (-4.09e4 - 2.97e4i)T^{2}$$
41 $$1 + (-84.5 - 260. i)T + (-5.57e4 + 4.05e4i)T^{2}$$
43 $$1 - 388.T + 7.95e4T^{2}$$
47 $$1 + (16.0 + 49.3i)T + (-8.39e4 + 6.10e4i)T^{2}$$
53 $$1 + (-333. + 242. i)T + (4.60e4 - 1.41e5i)T^{2}$$
59 $$1 + (-8.12 + 24.9i)T + (-1.66e5 - 1.20e5i)T^{2}$$
61 $$1 + (132. + 96.4i)T + (7.01e4 + 2.15e5i)T^{2}$$
67 $$1 - 276.T + 3.00e5T^{2}$$
71 $$1 + (-418. - 303. i)T + (1.10e5 + 3.40e5i)T^{2}$$
73 $$1 + (74.6 - 229. i)T + (-3.14e5 - 2.28e5i)T^{2}$$
79 $$1 + (-220. + 160. i)T + (1.52e5 - 4.68e5i)T^{2}$$
83 $$1 + (58.7 + 42.6i)T + (1.76e5 + 5.43e5i)T^{2}$$
89 $$1 + 1.19e3T + 7.04e5T^{2}$$
97 $$1 + (1.18e3 - 860. i)T + (2.82e5 - 8.68e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$