# Properties

 Degree 2 Conductor $3 \cdot 11$ Sign $0.957 - 0.288i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 1.65i)3-s − 2·4-s − 3.31i·5-s + (−2.5 + 1.65i)9-s + 3.31i·11-s + (−1 − 3.31i)12-s + (5.5 − 1.65i)15-s + 4·16-s + 6.63i·20-s − 3.31i·23-s − 6·25-s + (−4 − 3.31i)27-s + 5·31-s + (−5.5 + 1.65i)33-s + (5 − 3.31i)36-s − 7·37-s + ⋯
 L(s)  = 1 + (0.288 + 0.957i)3-s − 4-s − 1.48i·5-s + (−0.833 + 0.552i)9-s + 1.00i·11-s + (−0.288 − 0.957i)12-s + (1.42 − 0.428i)15-s + 16-s + 1.48i·20-s − 0.691i·23-s − 1.20·25-s + (−0.769 − 0.638i)27-s + 0.898·31-s + (−0.957 + 0.288i)33-s + (0.833 − 0.552i)36-s − 1.15·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$33$$    =    $$3 \cdot 11$$ $$\varepsilon$$ = $0.957 - 0.288i$ motivic weight = $$1$$ character : $\chi_{33} (32, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 33,\ (\ :1/2),\ 0.957 - 0.288i)$ $L(1)$ $\approx$ $0.660751 + 0.0974455i$ $L(\frac12)$ $\approx$ $0.660751 + 0.0974455i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;11\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-0.5 - 1.65i)T$$
11 $$1 - 3.31iT$$
good2 $$1 + 2T^{2}$$
5 $$1 + 3.31iT - 5T^{2}$$
7 $$1 - 7T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 3.31iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 5T + 31T^{2}$$
37 $$1 + 7T + 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 - 6.63iT - 47T^{2}$$
53 $$1 + 13.2iT - 53T^{2}$$
59 $$1 + 3.31iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 13T + 67T^{2}$$
71 $$1 - 16.5iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 16.5iT - 89T^{2}$$
97 $$1 - 17T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}