# Properties

 Degree 2 Conductor 11 Sign $0.964 - 0.262i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.67 + 1.19i)2-s + (−1.64 + 1.19i)3-s + (−0.850 − 0.618i)4-s + (−1.76 − 5.43i)5-s + (−7.47 + 2.43i)6-s + (−2.52 + 3.47i)7-s + (−38.7 − 53.3i)8-s + (−23.7 + 73.1i)9-s − 22.1i·10-s + (120. + 3.19i)11-s + 2.13·12-s + (189. + 61.5i)13-s + (−13.4 + 9.77i)14-s + (9.41 + 6.83i)15-s + (−73.5 − 226. i)16-s + (−269. + 87.5i)17-s + ⋯
 L(s)  = 1 + (0.919 + 0.298i)2-s + (−0.182 + 0.132i)3-s + (−0.0531 − 0.0386i)4-s + (−0.0706 − 0.217i)5-s + (−0.207 + 0.0675i)6-s + (−0.0515 + 0.0710i)7-s + (−0.605 − 0.833i)8-s + (−0.293 + 0.902i)9-s − 0.221i·10-s + (0.999 + 0.0264i)11-s + 0.0148·12-s + (1.12 + 0.364i)13-s + (−0.0686 + 0.0498i)14-s + (0.0418 + 0.0303i)15-s + (−0.287 − 0.884i)16-s + (−0.932 + 0.302i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$11$$ $$\varepsilon$$ = $0.964 - 0.262i$ motivic weight = $$4$$ character : $\chi_{11} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 11,\ (\ :2),\ 0.964 - 0.262i)$ $L(\frac{5}{2})$ $\approx$ $1.37220 + 0.183408i$ $L(\frac12)$ $\approx$ $1.37220 + 0.183408i$ $L(3)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $$F_p$$ is a polynomial of degree 2. If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 $$1 + (-120. - 3.19i)T$$
good2 $$1 + (-3.67 - 1.19i)T + (12.9 + 9.40i)T^{2}$$
3 $$1 + (1.64 - 1.19i)T + (25.0 - 77.0i)T^{2}$$
5 $$1 + (1.76 + 5.43i)T + (-505. + 367. i)T^{2}$$
7 $$1 + (2.52 - 3.47i)T + (-741. - 2.28e3i)T^{2}$$
13 $$1 + (-189. - 61.5i)T + (2.31e4 + 1.67e4i)T^{2}$$
17 $$1 + (269. - 87.5i)T + (6.75e4 - 4.90e4i)T^{2}$$
19 $$1 + (237. + 326. i)T + (-4.02e4 + 1.23e5i)T^{2}$$
23 $$1 + 189.T + 2.79e5T^{2}$$
29 $$1 + (797. - 1.09e3i)T + (-2.18e5 - 6.72e5i)T^{2}$$
31 $$1 + (-421. + 1.29e3i)T + (-7.47e5 - 5.42e5i)T^{2}$$
37 $$1 + (-690. - 501. i)T + (5.79e5 + 1.78e6i)T^{2}$$
41 $$1 + (-218. - 301. i)T + (-8.73e5 + 2.68e6i)T^{2}$$
43 $$1 - 854. iT - 3.41e6T^{2}$$
47 $$1 + (-1.28e3 + 932. i)T + (1.50e6 - 4.64e6i)T^{2}$$
53 $$1 + (274. - 846. i)T + (-6.38e6 - 4.63e6i)T^{2}$$
59 $$1 + (4.67e3 + 3.39e3i)T + (3.74e6 + 1.15e7i)T^{2}$$
61 $$1 + (3.71e3 - 1.20e3i)T + (1.12e7 - 8.13e6i)T^{2}$$
67 $$1 - 6.19e3T + 2.01e7T^{2}$$
71 $$1 + (-1.65e3 - 5.08e3i)T + (-2.05e7 + 1.49e7i)T^{2}$$
73 $$1 + (-3.57e3 + 4.91e3i)T + (-8.77e6 - 2.70e7i)T^{2}$$
79 $$1 + (4.60e3 + 1.49e3i)T + (3.15e7 + 2.28e7i)T^{2}$$
83 $$1 + (2.42e3 - 786. i)T + (3.83e7 - 2.78e7i)T^{2}$$
89 $$1 + 1.60e3T + 6.27e7T^{2}$$
97 $$1 + (-4.66e3 + 1.43e4i)T + (-7.16e7 - 5.20e7i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}