Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.985 - 0.172i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (3.63 + 4.33i)2-s + (−5.51 + 31.5i)4-s − 80.5i·5-s + 216. i·7-s + (−156. + 90.8i)8-s + (348. − 293. i)10-s + 153.·11-s − 945.·13-s + (−938. + 789. i)14-s + (−963. − 347. i)16-s + 2.18e3i·17-s + 719. i·19-s + (2.53e3 + 443. i)20-s + (558. + 664. i)22-s − 2.62e3·23-s + ⋯
 L(s)  = 1 + (0.643 + 0.765i)2-s + (−0.172 + 0.985i)4-s − 1.44i·5-s + 1.67i·7-s + (−0.864 + 0.501i)8-s + (1.10 − 0.926i)10-s + 0.382·11-s − 1.55·13-s + (−1.28 + 1.07i)14-s + (−0.940 − 0.339i)16-s + 1.83i·17-s + 0.457i·19-s + (1.41 + 0.248i)20-s + (0.245 + 0.292i)22-s − 1.03·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.985 - 0.172i$ motivic weight = $$5$$ character : $\chi_{108} (107, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ -0.985 - 0.172i)$$ $$L(3)$$ $$\approx$$ $$0.126991 + 1.46392i$$ $$L(\frac12)$$ $$\approx$$ $$0.126991 + 1.46392i$$ $$L(\frac{7}{2})$$ 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 + (-3.63 - 4.33i)T$$
3 $$1$$
good5 $$1 + 80.5iT - 3.12e3T^{2}$$
7 $$1 - 216. iT - 1.68e4T^{2}$$
11 $$1 - 153.T + 1.61e5T^{2}$$
13 $$1 + 945.T + 3.71e5T^{2}$$
17 $$1 - 2.18e3iT - 1.41e6T^{2}$$
19 $$1 - 719. iT - 2.47e6T^{2}$$
23 $$1 + 2.62e3T + 6.43e6T^{2}$$
29 $$1 - 155. iT - 2.05e7T^{2}$$
31 $$1 + 4.90e3iT - 2.86e7T^{2}$$
37 $$1 - 114.T + 6.93e7T^{2}$$
41 $$1 + 3.35e3iT - 1.15e8T^{2}$$
43 $$1 - 1.31e4iT - 1.47e8T^{2}$$
47 $$1 - 9.40e3T + 2.29e8T^{2}$$
53 $$1 - 1.96e4iT - 4.18e8T^{2}$$
59 $$1 - 2.15e4T + 7.14e8T^{2}$$
61 $$1 - 3.32e4T + 8.44e8T^{2}$$
67 $$1 - 2.34e4iT - 1.35e9T^{2}$$
71 $$1 - 2.40e3T + 1.80e9T^{2}$$
73 $$1 - 4.86e3T + 2.07e9T^{2}$$
79 $$1 - 1.19e4iT - 3.07e9T^{2}$$
83 $$1 - 8.39e4T + 3.93e9T^{2}$$
89 $$1 + 3.68e4iT - 5.58e9T^{2}$$
97 $$1 + 2.82e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}