# Properties

 Degree 2 Conductor 11 Sign $0.673 - 0.739i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.10 + 5.65i)2-s + (−4.15 − 12.7i)3-s + (−10.1 + 31.2i)4-s + (−10.0 − 7.30i)5-s + (55.2 − 75.9i)6-s + (23.8 + 7.73i)7-s + (−111. + 36.2i)8-s + (−80.7 + 58.6i)9-s − 86.8i·10-s + (16.1 + 119. i)11-s + 441.·12-s + (−110. − 151. i)13-s + (54.0 + 166. i)14-s + (−51.6 + 158. i)15-s + (−239. − 173. i)16-s + (−30.5 + 42.0i)17-s + ⋯
 L(s)  = 1 + (1.02 + 1.41i)2-s + (−0.461 − 1.42i)3-s + (−0.633 + 1.95i)4-s + (−0.402 − 0.292i)5-s + (1.53 − 2.11i)6-s + (0.485 + 0.157i)7-s + (−1.74 + 0.566i)8-s + (−0.996 + 0.724i)9-s − 0.868i·10-s + (0.133 + 0.991i)11-s + 3.06·12-s + (−0.653 − 0.898i)13-s + (0.275 + 0.848i)14-s + (−0.229 + 0.706i)15-s + (−0.933 − 0.678i)16-s + (−0.105 + 0.145i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$11$$ $$\varepsilon$$ = $0.673 - 0.739i$ motivic weight = $$4$$ character : $\chi_{11} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 11,\ (\ :2),\ 0.673 - 0.739i)$ $L(\frac{5}{2})$ $\approx$ $1.27894 + 0.565046i$ $L(\frac12)$ $\approx$ $1.27894 + 0.565046i$ $L(3)$ 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 + (-16.1 - 119. i)T$$
good2 $$1 + (-4.10 - 5.65i)T + (-4.94 + 15.2i)T^{2}$$
3 $$1 + (4.15 + 12.7i)T + (-65.5 + 47.6i)T^{2}$$
5 $$1 + (10.0 + 7.30i)T + (193. + 594. i)T^{2}$$
7 $$1 + (-23.8 - 7.73i)T + (1.94e3 + 1.41e3i)T^{2}$$
13 $$1 + (110. + 151. i)T + (-8.82e3 + 2.71e4i)T^{2}$$
17 $$1 + (30.5 - 42.0i)T + (-2.58e4 - 7.94e4i)T^{2}$$
19 $$1 + (-369. + 119. i)T + (1.05e5 - 7.66e4i)T^{2}$$
23 $$1 - 164.T + 2.79e5T^{2}$$
29 $$1 + (-596. - 193. i)T + (5.72e5 + 4.15e5i)T^{2}$$
31 $$1 + (974. - 708. i)T + (2.85e5 - 8.78e5i)T^{2}$$
37 $$1 + (-203. + 627. i)T + (-1.51e6 - 1.10e6i)T^{2}$$
41 $$1 + (2.18e3 - 710. i)T + (2.28e6 - 1.66e6i)T^{2}$$
43 $$1 + 1.74e3iT - 3.41e6T^{2}$$
47 $$1 + (-1.14e3 - 3.51e3i)T + (-3.94e6 + 2.86e6i)T^{2}$$
53 $$1 + (-63.9 + 46.4i)T + (2.43e6 - 7.50e6i)T^{2}$$
59 $$1 + (-341. + 1.04e3i)T + (-9.80e6 - 7.12e6i)T^{2}$$
61 $$1 + (-137. + 188. i)T + (-4.27e6 - 1.31e7i)T^{2}$$
67 $$1 - 6.39e3T + 2.01e7T^{2}$$
71 $$1 + (-4.46e3 - 3.24e3i)T + (7.85e6 + 2.41e7i)T^{2}$$
73 $$1 + (-2.97e3 - 968. i)T + (2.29e7 + 1.66e7i)T^{2}$$
79 $$1 + (1.85e3 + 2.55e3i)T + (-1.20e7 + 3.70e7i)T^{2}$$
83 $$1 + (-3.66e3 + 5.04e3i)T + (-1.46e7 - 4.51e7i)T^{2}$$
89 $$1 + 1.26e3T + 6.27e7T^{2}$$
97 $$1 + (-4.42e3 + 3.21e3i)T + (2.73e7 - 8.41e7i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}