# Properties

 Degree 2 Conductor $2 \cdot 11$ Sign $0.982 - 0.187i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.61 − 1.17i)2-s + (2.64 + 8.12i)3-s + (1.23 − 3.80i)4-s + (−10.3 − 7.48i)5-s + (13.8 + 10.0i)6-s + (7.24 − 22.3i)7-s + (−2.47 − 7.60i)8-s + (−37.2 + 27.0i)9-s − 25.4·10-s + (−3.69 + 36.2i)11-s + 34.1·12-s + (−9.27 + 6.73i)13-s + (−14.4 − 44.6i)14-s + (33.6 − 103. i)15-s + (−12.9 − 9.40i)16-s + (52.9 + 38.5i)17-s + ⋯
 L(s)  = 1 + (0.572 − 0.415i)2-s + (0.508 + 1.56i)3-s + (0.154 − 0.475i)4-s + (−0.921 − 0.669i)5-s + (0.940 + 0.683i)6-s + (0.391 − 1.20i)7-s + (−0.109 − 0.336i)8-s + (−1.37 + 1.00i)9-s − 0.805·10-s + (−0.101 + 0.994i)11-s + 0.822·12-s + (−0.197 + 0.143i)13-s + (−0.276 − 0.851i)14-s + (0.578 − 1.78i)15-s + (−0.202 − 0.146i)16-s + (0.756 + 0.549i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$22$$    =    $$2 \cdot 11$$ $$\varepsilon$$ = $0.982 - 0.187i$ motivic weight = $$3$$ character : $\chi_{22} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 22,\ (\ :3/2),\ 0.982 - 0.187i)$ $L(2)$ $\approx$ $1.46851 + 0.139106i$ $L(\frac12)$ $\approx$ $1.46851 + 0.139106i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;11\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (-1.61 + 1.17i)T$$
11 $$1 + (3.69 - 36.2i)T$$
good3 $$1 + (-2.64 - 8.12i)T + (-21.8 + 15.8i)T^{2}$$
5 $$1 + (10.3 + 7.48i)T + (38.6 + 118. i)T^{2}$$
7 $$1 + (-7.24 + 22.3i)T + (-277. - 201. i)T^{2}$$
13 $$1 + (9.27 - 6.73i)T + (678. - 2.08e3i)T^{2}$$
17 $$1 + (-52.9 - 38.5i)T + (1.51e3 + 4.67e3i)T^{2}$$
19 $$1 + (2.24 + 6.89i)T + (-5.54e3 + 4.03e3i)T^{2}$$
23 $$1 + 104.T + 1.21e4T^{2}$$
29 $$1 + (39.3 - 121. i)T + (-1.97e4 - 1.43e4i)T^{2}$$
31 $$1 + (-233. + 169. i)T + (9.20e3 - 2.83e4i)T^{2}$$
37 $$1 + (26.3 - 81.2i)T + (-4.09e4 - 2.97e4i)T^{2}$$
41 $$1 + (41.8 + 128. i)T + (-5.57e4 + 4.05e4i)T^{2}$$
43 $$1 - 353.T + 7.95e4T^{2}$$
47 $$1 + (41.5 + 128. i)T + (-8.39e4 + 6.10e4i)T^{2}$$
53 $$1 + (405. - 294. i)T + (4.60e4 - 1.41e5i)T^{2}$$
59 $$1 + (-201. + 619. i)T + (-1.66e5 - 1.20e5i)T^{2}$$
61 $$1 + (295. + 215. i)T + (7.01e4 + 2.15e5i)T^{2}$$
67 $$1 + 294.T + 3.00e5T^{2}$$
71 $$1 + (107. + 77.8i)T + (1.10e5 + 3.40e5i)T^{2}$$
73 $$1 + (-145. + 446. i)T + (-3.14e5 - 2.28e5i)T^{2}$$
79 $$1 + (330. - 239. i)T + (1.52e5 - 4.68e5i)T^{2}$$
83 $$1 + (-1.09e3 - 799. i)T + (1.76e5 + 5.43e5i)T^{2}$$
89 $$1 + 260.T + 7.04e5T^{2}$$
97 $$1 + (1.14e3 - 831. i)T + (2.82e5 - 8.68e5i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}