# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5$ Sign $0.894 + 0.447i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + i·3-s − 4-s + (−2 − i)5-s + 6-s + 2i·7-s + i·8-s − 9-s + (−1 + 2i)10-s + 2·11-s − i·12-s − 6i·13-s + 2·14-s + (1 − 2i)15-s + 16-s + 2i·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 1.66i·13-s + 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 0.485i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$30$$    =    $$2 \cdot 3 \cdot 5$$ $$\varepsilon$$ = $0.894 + 0.447i$ motivic weight = $$1$$ character : $\chi_{30} (19, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 30,\ (\ :1/2),\ 0.894 + 0.447i)$ $L(1)$ $\approx$ $0.612627 - 0.144621i$ $L(\frac12)$ $\approx$ $0.612627 - 0.144621i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + iT$$
3 $$1 - iT$$
5 $$1 + (2 + i)T$$
good7 $$1 - 2iT - 7T^{2}$$
11 $$1 - 2T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 8T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 10T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + 8iT - 67T^{2}$$
71 $$1 - 12T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 4iT - 83T^{2}$$
89 $$1 - 10T + 89T^{2}$$
97 $$1 + 8iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}