# Properties

 Degree $2$ Conductor $33$ Sign $-1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 9·3-s − 31·4-s − 92·5-s + 9·6-s − 26·7-s − 63·8-s + 81·9-s − 92·10-s + 121·11-s − 279·12-s − 692·13-s − 26·14-s − 828·15-s + 929·16-s − 1.44e3·17-s + 81·18-s + 2.16e3·19-s + 2.85e3·20-s − 234·21-s + 121·22-s − 1.58e3·23-s − 567·24-s + 5.33e3·25-s − 692·26-s + 729·27-s + 806·28-s + ⋯
 L(s)  = 1 + 0.176·2-s + 0.577·3-s − 0.968·4-s − 1.64·5-s + 0.102·6-s − 0.200·7-s − 0.348·8-s + 1/3·9-s − 0.290·10-s + 0.301·11-s − 0.559·12-s − 1.13·13-s − 0.0354·14-s − 0.950·15-s + 0.907·16-s − 1.21·17-s + 0.0589·18-s + 1.37·19-s + 1.59·20-s − 0.115·21-s + 0.0533·22-s − 0.623·23-s − 0.200·24-s + 1.70·25-s − 0.200·26-s + 0.192·27-s + 0.194·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $-1$ Motivic weight: $$5$$ Character: $\chi_{33} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 33,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - p^{2} T$$
11 $$1 - p^{2} T$$
good2 $$1 - T + p^{5} T^{2}$$
5 $$1 + 92 T + p^{5} T^{2}$$
7 $$1 + 26 T + p^{5} T^{2}$$
13 $$1 + 692 T + p^{5} T^{2}$$
17 $$1 + 1442 T + p^{5} T^{2}$$
19 $$1 - 2160 T + p^{5} T^{2}$$
23 $$1 + 1582 T + p^{5} T^{2}$$
29 $$1 + 5526 T + p^{5} T^{2}$$
31 $$1 - 4792 T + p^{5} T^{2}$$
37 $$1 + 10194 T + p^{5} T^{2}$$
41 $$1 + 10622 T + p^{5} T^{2}$$
43 $$1 - 8580 T + p^{5} T^{2}$$
47 $$1 + 2362 T + p^{5} T^{2}$$
53 $$1 + 30804 T + p^{5} T^{2}$$
59 $$1 - 6416 T + p^{5} T^{2}$$
61 $$1 - 42096 T + p^{5} T^{2}$$
67 $$1 + 28444 T + p^{5} T^{2}$$
71 $$1 - 45690 T + p^{5} T^{2}$$
73 $$1 + 18374 T + p^{5} T^{2}$$
79 $$1 + 105214 T + p^{5} T^{2}$$
83 $$1 - 62292 T + p^{5} T^{2}$$
89 $$1 + 72246 T + p^{5} T^{2}$$
97 $$1 - 79262 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$