# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.479 + 0.877i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.988 + 3.87i)2-s + (−14.0 + 7.66i)4-s − 20.7·5-s − 5.38i·7-s + (−43.5 − 46.8i)8-s + (−20.5 − 80.4i)10-s − 115. i·11-s + 207.·13-s + (20.8 − 5.32i)14-s + (138. − 215. i)16-s − 383.·17-s − 618. i·19-s + (291. − 159. i)20-s + (448. − 114. i)22-s + 82.9i·23-s + ⋯
 L(s)  = 1 + (0.247 + 0.968i)2-s + (−0.877 + 0.479i)4-s − 0.830·5-s − 0.109i·7-s + (−0.681 − 0.732i)8-s + (−0.205 − 0.804i)10-s − 0.955i·11-s + 1.22·13-s + (0.106 − 0.0271i)14-s + (0.540 − 0.841i)16-s − 1.32·17-s − 1.71i·19-s + (0.728 − 0.397i)20-s + (0.925 − 0.236i)22-s + 0.156i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.479 + 0.877i$ motivic weight = $$4$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.479 + 0.877i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.593095 - 0.351981i$$ $$L(\frac12)$$ $$\approx$$ $$0.593095 - 0.351981i$$ $$L(3)$$ 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 + (-0.988 - 3.87i)T$$
3 $$1$$
good5 $$1 + 20.7T + 625T^{2}$$
7 $$1 + 5.38iT - 2.40e3T^{2}$$
11 $$1 + 115. iT - 1.46e4T^{2}$$
13 $$1 - 207.T + 2.85e4T^{2}$$
17 $$1 + 383.T + 8.35e4T^{2}$$
19 $$1 + 618. iT - 1.30e5T^{2}$$
23 $$1 - 82.9iT - 2.79e5T^{2}$$
29 $$1 + 201.T + 7.07e5T^{2}$$
31 $$1 + 196. iT - 9.23e5T^{2}$$
37 $$1 - 340.T + 1.87e6T^{2}$$
41 $$1 + 2.79e3T + 2.82e6T^{2}$$
43 $$1 + 254. iT - 3.41e6T^{2}$$
47 $$1 + 2.25e3iT - 4.87e6T^{2}$$
53 $$1 + 4.11e3T + 7.89e6T^{2}$$
59 $$1 - 1.59e3iT - 1.21e7T^{2}$$
61 $$1 + 6.08e3T + 1.38e7T^{2}$$
67 $$1 - 7.38e3iT - 2.01e7T^{2}$$
71 $$1 + 9.34e3iT - 2.54e7T^{2}$$
73 $$1 + 5.32e3T + 2.83e7T^{2}$$
79 $$1 + 5.97e3iT - 3.89e7T^{2}$$
83 $$1 - 1.00e4iT - 4.74e7T^{2}$$
89 $$1 - 1.36e4T + 6.27e7T^{2}$$
97 $$1 - 2.17e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}