# Properties

 Degree 2 Conductor $3^{3} \cdot 7$ Sign $0.574 - 0.818i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.730 + 1.26i)2-s + (−0.0665 + 0.115i)4-s + (0.296 + 0.514i)5-s + (2.32 − 1.26i)7-s + 2.72·8-s + (−0.433 + 0.750i)10-s + (−2.23 + 3.86i)11-s − 4.51·13-s + (3.29 + 2.01i)14-s + (2.12 + 3.67i)16-s + (0.136 − 0.236i)17-s + (−1.43 − 2.48i)19-s − 0.0789·20-s − 6.51·22-s + (−2.52 − 4.37i)23-s + ⋯
 L(s)  = 1 + (0.516 + 0.894i)2-s + (−0.0332 + 0.0576i)4-s + (0.132 + 0.229i)5-s + (0.878 − 0.478i)7-s + 0.964·8-s + (−0.137 + 0.237i)10-s + (−0.672 + 1.16i)11-s − 1.25·13-s + (0.881 + 0.538i)14-s + (0.531 + 0.919i)16-s + (0.0331 − 0.0574i)17-s + (−0.328 − 0.569i)19-s − 0.0176·20-s − 1.38·22-s + (−0.526 − 0.912i)23-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$189$$    =    $$3^{3} \cdot 7$$ $$\varepsilon$$ = $0.574 - 0.818i$ motivic weight = $$1$$ character : $\chi_{189} (109, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 189,\ (\ :1/2),\ 0.574 - 0.818i)$ $L(1)$ $\approx$ $1.48105 + 0.769802i$ $L(\frac12)$ $\approx$ $1.48105 + 0.769802i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1$$
7 $$1 + (-2.32 + 1.26i)T$$
good2 $$1 + (-0.730 - 1.26i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-0.296 - 0.514i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (2.23 - 3.86i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 4.51T + 13T^{2}$$
17 $$1 + (-0.136 + 0.236i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.43 + 2.48i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (2.52 + 4.37i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 0.352T + 29T^{2}$$
31 $$1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-3.32 - 5.75i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 10.8T + 41T^{2}$$
43 $$1 - 3.38T + 43T^{2}$$
47 $$1 + (6.21 + 10.7i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (5.66 - 9.80i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-1.36 - 2.36i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 2.60T + 71T^{2}$$
73 $$1 + (5.55 - 9.62i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (5.58 + 9.66i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 16.5T + 83T^{2}$$
89 $$1 + (-2.68 - 4.65i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 2.26T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}