# Properties

 Degree 2 Conductor 61 Sign $-0.855 + 0.518i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 + 0.866i)2-s − 2·3-s + (0.5 − 0.866i)4-s + (−1.5 − 2.59i)5-s + (3 − 1.73i)6-s + (−3 + 1.73i)7-s − 1.73i·8-s + 9-s + (4.5 + 2.59i)10-s + 3.46i·11-s + (−1 + 1.73i)12-s + (1 + 1.73i)13-s + (3 − 5.19i)14-s + (3 + 5.19i)15-s + (2.49 + 4.33i)16-s + (−6 − 3.46i)17-s + ⋯
 L(s)  = 1 + (−1.06 + 0.612i)2-s − 1.15·3-s + (0.250 − 0.433i)4-s + (−0.670 − 1.16i)5-s + (1.22 − 0.707i)6-s + (−1.13 + 0.654i)7-s − 0.612i·8-s + 0.333·9-s + (1.42 + 0.821i)10-s + 1.04i·11-s + (−0.288 + 0.500i)12-s + (0.277 + 0.480i)13-s + (0.801 − 1.38i)14-s + (0.774 + 1.34i)15-s + (0.624 + 1.08i)16-s + (−1.45 − 0.840i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$61$$ $$\varepsilon$$ = $-0.855 + 0.518i$ motivic weight = $$1$$ character : $\chi_{61} (14, \cdot )$ primitive : yes self-dual : no analytic rank = 1 Selberg data = $(2,\ 61,\ (\ :1/2),\ -0.855 + 0.518i)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 61$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 61$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad61 $$1 + (0.5 + 7.79i)T$$
good2 $$1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2}$$
3 $$1 + 2T + 3T^{2}$$
5 $$1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 - 3.46iT - 11T^{2}$$
13 $$1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (9 + 5.19i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 1.73iT - 37T^{2}$$
41 $$1 - 3T + 41T^{2}$$
43 $$1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 5.19iT - 53T^{2}$$
59 $$1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2}$$
67 $$1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (9 - 5.19i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 15.5iT - 89T^{2}$$
97 $$1 + (0.5 - 0.866i)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}