Properties

 Degree 2 Conductor $3 \cdot 19$ Sign $0.813 + 0.582i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + (0.499 − 0.866i)6-s + 7-s − 3·8-s + (−0.499 + 0.866i)9-s − 2·11-s + 12-s + (−2.5 + 4.33i)13-s + (−0.5 − 0.866i)14-s + (0.500 + 0.866i)16-s + (2 + 3.46i)17-s + 0.999·18-s + (−4 − 1.73i)19-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.204 − 0.353i)6-s + 0.377·7-s − 1.06·8-s + (−0.166 + 0.288i)9-s − 0.603·11-s + 0.288·12-s + (−0.693 + 1.20i)13-s + (−0.133 − 0.231i)14-s + (0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + 0.235·18-s + (−0.917 − 0.397i)19-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$57$$    =    $$3 \cdot 19$$ $$\varepsilon$$ = $0.813 + 0.582i$ motivic weight = $$1$$ character : $\chi_{57} (7, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 57,\ (\ :1/2),\ 0.813 + 0.582i)$ $L(1)$ $\approx$ $0.775770 - 0.249138i$ $L(\frac12)$ $\approx$ $0.775770 - 0.249138i$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;19\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (-0.5 - 0.866i)T$$
19 $$1 + (4 + 1.73i)T$$
good2 $$1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-2.5 + 4.33i)T^{2}$$
7 $$1 - T + 7T^{2}$$
11 $$1 + 2T + 11T^{2}$$
13 $$1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2}$$
23 $$1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + 3T + 31T^{2}$$
37 $$1 - 3T + 37T^{2}$$
41 $$1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (1 + 1.73i)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}