# Properties

 Degree 2 Conductor 11 Sign $0.0909 + 0.995i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.47i·2-s − 3·3-s − 14·4-s + 31·5-s + 16.4i·6-s + 54.7i·7-s − 10.9i·8-s − 72·9-s − 169. i·10-s + (11 + 120. i)11-s + 42·12-s − 186. i·13-s + 300·14-s − 93·15-s − 284·16-s + 230. i·17-s + ⋯
 L(s)  = 1 − 1.36i·2-s − 0.333·3-s − 0.875·4-s + 1.23·5-s + 0.456i·6-s + 1.11i·7-s − 0.171i·8-s − 0.888·9-s − 1.69i·10-s + (0.0909 + 0.995i)11-s + 0.291·12-s − 1.10i·13-s + 1.53·14-s − 0.413·15-s − 1.10·16-s + 0.795i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0909 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$11$$ $$\varepsilon$$ = $0.0909 + 0.995i$ motivic weight = $$4$$ character : $\chi_{11} (10, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 11,\ (\ :2),\ 0.0909 + 0.995i)$ $L(\frac{5}{2})$ $\approx$ $0.794643 - 0.725407i$ $L(\frac12)$ $\approx$ $0.794643 - 0.725407i$ $L(3)$ 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 + (-11 - 120. i)T$$
good2 $$1 + 5.47iT - 16T^{2}$$
3 $$1 + 3T + 81T^{2}$$
5 $$1 - 31T + 625T^{2}$$
7 $$1 - 54.7iT - 2.40e3T^{2}$$
13 $$1 + 186. iT - 2.85e4T^{2}$$
17 $$1 - 230. iT - 8.35e4T^{2}$$
19 $$1 + 98.5iT - 1.30e5T^{2}$$
23 $$1 - 277T + 2.79e5T^{2}$$
29 $$1 + 1.27e3iT - 7.07e5T^{2}$$
31 $$1 + 1.36e3T + 9.23e5T^{2}$$
37 $$1 - 167T + 1.87e6T^{2}$$
41 $$1 + 1.06e3iT - 2.82e6T^{2}$$
43 $$1 - 1.20e3iT - 3.41e6T^{2}$$
47 $$1 - 1.70e3T + 4.87e6T^{2}$$
53 $$1 - 4.52e3T + 7.89e6T^{2}$$
59 $$1 + 2.36e3T + 1.21e7T^{2}$$
61 $$1 - 3.96e3iT - 1.38e7T^{2}$$
67 $$1 + 2.80e3T + 2.01e7T^{2}$$
71 $$1 - 3.39e3T + 2.54e7T^{2}$$
73 $$1 + 3.31e3iT - 2.83e7T^{2}$$
79 $$1 + 6.09e3iT - 3.89e7T^{2}$$
83 $$1 - 832. iT - 4.74e7T^{2}$$
89 $$1 + 4.67e3T + 6.27e7T^{2}$$
97 $$1 - 4.24e3T + 8.85e7T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}