Properties

 Degree 2 Conductor $5 \cdot 31$ Sign $-0.5 - 0.866i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + 1.73i·2-s − 1.99·4-s + (0.5 + 0.866i)5-s − 1.73i·7-s − 1.73i·8-s − 9-s + (−1.49 + 0.866i)10-s + 2.99·14-s + 0.999·16-s − 1.73i·18-s + 19-s + (−0.999 − 1.73i)20-s + (−0.499 + 0.866i)25-s + 3.46i·28-s − 31-s + ⋯
 L(s)  = 1 + 1.73i·2-s − 1.99·4-s + (0.5 + 0.866i)5-s − 1.73i·7-s − 1.73i·8-s − 9-s + (−1.49 + 0.866i)10-s + 2.99·14-s + 0.999·16-s − 1.73i·18-s + 19-s + (−0.999 − 1.73i)20-s + (−0.499 + 0.866i)25-s + 3.46i·28-s − 31-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$155$$    =    $$5 \cdot 31$$ $$\varepsilon$$ = $-0.5 - 0.866i$ motivic weight = $$0$$ character : $\chi_{155} (154, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 155,\ (\ :0),\ -0.5 - 0.866i)$ $L(\frac{1}{2})$ $\approx$ $0.6078176341$ $L(\frac12)$ $\approx$ $0.6078176341$ $L(1)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;31\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{5,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1 + (-0.5 - 0.866i)T$$
31 $$1 + T$$
good2 $$1 - 1.73iT - T^{2}$$
3 $$1 + T^{2}$$
7 $$1 + 1.73iT - T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + T^{2}$$
19 $$1 - T + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 + T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + T + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - 1.73iT - T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}