# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.850 + 0.526i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.72 + 0.773i)2-s + (6.80 + 4.21i)4-s − 20.8i·5-s − 13.9i·7-s + (15.2 + 16.7i)8-s + (16.1 − 56.7i)10-s + 34.5·11-s − 31.3·13-s + (10.7 − 37.8i)14-s + (28.5 + 57.2i)16-s + 34.4i·17-s + 120. i·19-s + (87.7 − 141. i)20-s + (93.8 + 26.7i)22-s + 137.·23-s + ⋯
 L(s)  = 1 + (0.961 + 0.273i)2-s + (0.850 + 0.526i)4-s − 1.86i·5-s − 0.750i·7-s + (0.673 + 0.738i)8-s + (0.510 − 1.79i)10-s + 0.945·11-s − 0.668·13-s + (0.205 − 0.722i)14-s + (0.445 + 0.895i)16-s + 0.491i·17-s + 1.45i·19-s + (0.981 − 1.58i)20-s + (0.909 + 0.258i)22-s + 1.24·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.850 + 0.526i$ motivic weight = $$3$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ 0.850 + 0.526i)$$ $$L(2)$$ $$\approx$$ $$2.70750 - 0.770263i$$ $$L(\frac12)$$ $$\approx$$ $$2.70750 - 0.770263i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-2.72 - 0.773i)T$$
3 $$1$$
good5 $$1 + 20.8iT - 125T^{2}$$
7 $$1 + 13.9iT - 343T^{2}$$
11 $$1 - 34.5T + 1.33e3T^{2}$$
13 $$1 + 31.3T + 2.19e3T^{2}$$
17 $$1 - 34.4iT - 4.91e3T^{2}$$
19 $$1 - 120. iT - 6.85e3T^{2}$$
23 $$1 - 137.T + 1.21e4T^{2}$$
29 $$1 + 93.1iT - 2.43e4T^{2}$$
31 $$1 + 111. iT - 2.97e4T^{2}$$
37 $$1 + 146.T + 5.06e4T^{2}$$
41 $$1 + 8.44iT - 6.89e4T^{2}$$
43 $$1 - 427. iT - 7.95e4T^{2}$$
47 $$1 + 318.T + 1.03e5T^{2}$$
53 $$1 - 291. iT - 1.48e5T^{2}$$
59 $$1 - 364.T + 2.05e5T^{2}$$
61 $$1 + 289.T + 2.26e5T^{2}$$
67 $$1 - 305. iT - 3.00e5T^{2}$$
71 $$1 + 102.T + 3.57e5T^{2}$$
73 $$1 - 442.T + 3.89e5T^{2}$$
79 $$1 - 245. iT - 4.93e5T^{2}$$
83 $$1 + 478.T + 5.71e5T^{2}$$
89 $$1 + 1.41e3iT - 7.04e5T^{2}$$
97 $$1 - 1.15e3T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}