# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.955 - 0.293i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.419 − 2.79i)2-s + (−7.64 − 2.34i)4-s + 1.49i·5-s − 26.1i·7-s + (−9.78 + 20.4i)8-s + (4.18 + 0.627i)10-s − 56.3·11-s − 41.3·13-s + (−73.2 − 10.9i)14-s + (52.9 + 35.9i)16-s + 51.0i·17-s − 79.0i·19-s + (3.51 − 11.4i)20-s + (−23.6 + 157. i)22-s − 27.3·23-s + ⋯
 L(s)  = 1 + (0.148 − 0.988i)2-s + (−0.955 − 0.293i)4-s + 0.133i·5-s − 1.41i·7-s + (−0.432 + 0.901i)8-s + (0.132 + 0.0198i)10-s − 1.54·11-s − 0.881·13-s + (−1.39 − 0.209i)14-s + (0.827 + 0.561i)16-s + 0.728i·17-s − 0.954i·19-s + (0.0392 − 0.127i)20-s + (−0.229 + 1.52i)22-s − 0.248·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.955 - 0.293i$ motivic weight = $$3$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -0.955 - 0.293i)$$ $$L(2)$$ $$\approx$$ $$0.117878 + 0.785220i$$ $$L(\frac12)$$ $$\approx$$ $$0.117878 + 0.785220i$$ $$L(\frac{5}{2})$$ 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 + (-0.419 + 2.79i)T$$
3 $$1$$
good5 $$1 - 1.49iT - 125T^{2}$$
7 $$1 + 26.1iT - 343T^{2}$$
11 $$1 + 56.3T + 1.33e3T^{2}$$
13 $$1 + 41.3T + 2.19e3T^{2}$$
17 $$1 - 51.0iT - 4.91e3T^{2}$$
19 $$1 + 79.0iT - 6.85e3T^{2}$$
23 $$1 + 27.3T + 1.21e4T^{2}$$
29 $$1 + 134. iT - 2.43e4T^{2}$$
31 $$1 + 187. iT - 2.97e4T^{2}$$
37 $$1 + 196.T + 5.06e4T^{2}$$
41 $$1 - 298. iT - 6.89e4T^{2}$$
43 $$1 + 465. iT - 7.95e4T^{2}$$
47 $$1 - 373.T + 1.03e5T^{2}$$
53 $$1 + 620. iT - 1.48e5T^{2}$$
59 $$1 - 321.T + 2.05e5T^{2}$$
61 $$1 - 674.T + 2.26e5T^{2}$$
67 $$1 - 576. iT - 3.00e5T^{2}$$
71 $$1 + 223.T + 3.57e5T^{2}$$
73 $$1 - 70.1T + 3.89e5T^{2}$$
79 $$1 - 1.05e3iT - 4.93e5T^{2}$$
83 $$1 + 1.21e3T + 5.71e5T^{2}$$
89 $$1 + 1.34e3iT - 7.04e5T^{2}$$
97 $$1 + 576.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}