# Properties

 Degree 2 Conductor $5 \cdot 11$ Sign $0.983 - 0.180i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.15 + 1.15i)3-s − 2i·4-s + (−1.65 + 1.5i)5-s − 0.316i·9-s − 3.31·11-s + (2.31 − 2.31i)12-s + (−3.65 − 0.183i)15-s − 4·16-s + (3 + 3.31i)20-s + (6.15 + 6.15i)23-s + (0.5 − 4.97i)25-s + (3.84 − 3.84i)27-s + 9.94·31-s + (−3.84 − 3.84i)33-s − 0.633·36-s + (−8.47 + 8.47i)37-s + ⋯
 L(s)  = 1 + (0.668 + 0.668i)3-s − i·4-s + (−0.741 + 0.670i)5-s − 0.105i·9-s − 1.00·11-s + (0.668 − 0.668i)12-s + (−0.944 − 0.0473i)15-s − 16-s + (0.670 + 0.741i)20-s + (1.28 + 1.28i)23-s + (0.100 − 0.994i)25-s + (0.739 − 0.739i)27-s + 1.78·31-s + (−0.668 − 0.668i)33-s − 0.105·36-s + (−1.39 + 1.39i)37-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$55$$    =    $$5 \cdot 11$$ $$\varepsilon$$ = $0.983 - 0.180i$ motivic weight = $$1$$ character : $\chi_{55} (43, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 55,\ (\ :1/2),\ 0.983 - 0.180i)$ $L(1)$ $\approx$ $0.904114 + 0.0823837i$ $L(\frac12)$ $\approx$ $0.904114 + 0.0823837i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;11\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 + (1.65 - 1.5i)T$$
11 $$1 + 3.31T$$
good2 $$1 + 2iT^{2}$$
3 $$1 + (-1.15 - 1.15i)T + 3iT^{2}$$
7 $$1 + 7iT^{2}$$
13 $$1 - 13iT^{2}$$
17 $$1 + 17iT^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + (-6.15 - 6.15i)T + 23iT^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 9.94T + 31T^{2}$$
37 $$1 + (8.47 - 8.47i)T - 37iT^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 43iT^{2}$$
47 $$1 + (2.68 - 2.68i)T - 47iT^{2}$$
53 $$1 + (9.63 + 9.63i)T + 53iT^{2}$$
59 $$1 + 3.31iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + (-1.52 + 1.52i)T - 67iT^{2}$$
71 $$1 + 3T + 71T^{2}$$
73 $$1 - 73iT^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 83iT^{2}$$
89 $$1 + 9iT - 89T^{2}$$
97 $$1 + (13.4 - 13.4i)T - 97iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}