# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.612 - 0.790i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.93 + 2.06i)2-s + (−0.538 + 7.98i)4-s + (4.71 + 2.72i)5-s + (−20.9 + 12.0i)7-s + (−17.5 + 14.3i)8-s + (3.48 + 14.9i)10-s + (25.3 + 43.9i)11-s + (25.0 − 43.4i)13-s + (−65.4 − 19.9i)14-s + (−63.4 − 8.60i)16-s + 51.7i·17-s + 27.9i·19-s + (−24.2 + 36.1i)20-s + (−41.8 + 137. i)22-s + (−3.93 + 6.81i)23-s + ⋯
 L(s)  = 1 + (0.682 + 0.730i)2-s + (−0.0673 + 0.997i)4-s + (0.421 + 0.243i)5-s + (−1.13 + 0.652i)7-s + (−0.774 + 0.632i)8-s + (0.110 + 0.474i)10-s + (0.696 + 1.20i)11-s + (0.535 − 0.927i)13-s + (−1.24 − 0.380i)14-s + (−0.990 − 0.134i)16-s + 0.737i·17-s + 0.337i·19-s + (−0.271 + 0.404i)20-s + (−0.405 + 1.33i)22-s + (−0.0356 + 0.0617i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.612 - 0.790i$ motivic weight = $$3$$ character : $\chi_{108} (71, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -0.612 - 0.790i)$$ $$L(2)$$ $$\approx$$ $$0.890455 + 1.81547i$$ $$L(\frac12)$$ $$\approx$$ $$0.890455 + 1.81547i$$ $$L(\frac{5}{2})$$ 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 + (-1.93 - 2.06i)T$$
3 $$1$$
good5 $$1 + (-4.71 - 2.72i)T + (62.5 + 108. i)T^{2}$$
7 $$1 + (20.9 - 12.0i)T + (171.5 - 297. i)T^{2}$$
11 $$1 + (-25.3 - 43.9i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-25.0 + 43.4i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 51.7iT - 4.91e3T^{2}$$
19 $$1 - 27.9iT - 6.85e3T^{2}$$
23 $$1 + (3.93 - 6.81i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-212. + 122. i)T + (1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-51.4 - 29.6i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 295.T + 5.06e4T^{2}$$
41 $$1 + (-146. - 84.7i)T + (3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-284. + 164. i)T + (3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (47.9 + 83.0i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 - 300. iT - 1.48e5T^{2}$$
59 $$1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-173. - 300. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (904. + 522. i)T + (1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 243.T + 3.57e5T^{2}$$
73 $$1 + 1.09e3T + 3.89e5T^{2}$$
79 $$1 + (-530. + 306. i)T + (2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-283. - 490. i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 + 212. iT - 7.04e5T^{2}$$
97 $$1 + (-234. - 405. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}