# Properties

 Degree $2$ Conductor $51$ Sign $0.970 + 0.242i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − i·3-s − 4-s − i·6-s + 4i·7-s − 3·8-s − 9-s − 4i·11-s + i·12-s + 2·13-s + 4i·14-s − 16-s + (1 − 4i)17-s − 18-s − 4·19-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577i·3-s − 0.5·4-s − 0.408i·6-s + 1.51i·7-s − 1.06·8-s − 0.333·9-s − 1.20i·11-s + 0.288i·12-s + 0.554·13-s + 1.06i·14-s − 0.250·16-s + (0.242 − 0.970i)17-s − 0.235·18-s − 0.917·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$51$$    =    $$3 \cdot 17$$ Sign: $0.970 + 0.242i$ Motivic weight: $$1$$ Character: $\chi_{51} (16, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 51,\ (\ :1/2),\ 0.970 + 0.242i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.977239 - 0.120303i$$ $$L(\frac12)$$ $$\approx$$ $$0.977239 - 0.120303i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + iT$$
17 $$1 + (-1 + 4i)T$$
good2 $$1 - T + 2T^{2}$$
5 $$1 - 5T^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 + 4iT - 11T^{2}$$
13 $$1 - 2T + 13T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 - 4iT - 31T^{2}$$
37 $$1 + 8iT - 37T^{2}$$
41 $$1 - 8iT - 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 - 6T + 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 8iT - 61T^{2}$$
67 $$1 - 12T + 67T^{2}$$
71 $$1 + 12iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 4iT - 79T^{2}$$
83 $$1 - 12T + 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 16iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$