# Properties

 Degree 2 Conductor $2^{2} \cdot 31$ Sign $-0.390 - 0.920i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
 L(s)  = 1 + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$124$$    =    $$2^{2} \cdot 31$$ $$\varepsilon$$ = $-0.390 - 0.920i$ motivic weight = $$0$$ character : $\chi_{124} (87, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 124,\ (\ :0),\ -0.390 - 0.920i)$ $L(\frac{1}{2})$ $\approx$ $0.4496221474$ $L(\frac12)$ $\approx$ $0.4496221474$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;31\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - iT$$
31 $$1 + iT$$
good3 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
5 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
7 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
11 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
13 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
17 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
19 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
37 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
43 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
71 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
73 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
83 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}