# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 7$ Sign $-0.281 - 0.959i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + (0.866 − 1.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−1.5 − 0.866i)6-s + (−2.5 + 0.866i)7-s + i·8-s + (−1.5 − 2.59i)9-s + (0.866 + 0.5i)10-s + (−4.09 + 2.36i)11-s + (−0.866 + 1.5i)12-s + (−3 + 1.73i)13-s + (0.866 + 2.5i)14-s + (0.866 + 1.5i)15-s + 16-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (0.499 − 0.866i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.612 − 0.353i)6-s + (−0.944 + 0.327i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.273 + 0.158i)10-s + (−1.23 + 0.713i)11-s + (−0.249 + 0.433i)12-s + (−0.832 + 0.480i)13-s + (0.231 + 0.668i)14-s + (0.223 + 0.387i)15-s + 0.250·16-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$630$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.281 - 0.959i$ motivic weight = $$1$$ character : $\chi_{630} (101, \cdot )$ primitive : yes self-dual : no analytic rank = $$1$$ Selberg data = $$(2,\ 630,\ (\ :1/2),\ -0.281 - 0.959i)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + (-0.866 + 1.5i)T$$
5 $$1 + (0.5 - 0.866i)T$$
7 $$1 + (2.5 - 0.866i)T$$
good11 $$1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-2.19 - 1.26i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (5.59 + 3.23i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + 8.19iT - 31T^{2}$$
37 $$1 + (3.09 + 5.36i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (1.59 - 2.76i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 9T + 47T^{2}$$
53 $$1 + (-6.29 - 3.63i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 - 2.19T + 59T^{2}$$
61 $$1 + 12.9iT - 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 - 4.73iT - 71T^{2}$$
73 $$1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 - 14.5T + 79T^{2}$$
83 $$1 + (5.59 - 9.69i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + (-2.19 - 3.80i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (7.39 + 4.26i)T + (48.5 + 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}