Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $0.852 - 0.522i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−3.49 − 1.95i)2-s + (8.36 + 13.6i)4-s − 47.1·5-s − 67.4i·7-s + (−2.56 − 63.9i)8-s + (164. + 92.0i)10-s + 69.3i·11-s + 3.28·13-s + (−131. + 235. i)14-s + (−115. + 228. i)16-s + 116.·17-s + 513. i·19-s + (−394. − 642. i)20-s + (135. − 242. i)22-s + 134. i·23-s + ⋯
 L(s)  = 1 + (−0.872 − 0.488i)2-s + (0.522 + 0.852i)4-s − 1.88·5-s − 1.37i·7-s + (−0.0400 − 0.999i)8-s + (1.64 + 0.920i)10-s + 0.573i·11-s + 0.0194·13-s + (−0.671 + 1.20i)14-s + (−0.453 + 0.891i)16-s + 0.404·17-s + 1.42i·19-s + (−0.985 − 1.60i)20-s + (0.280 − 0.500i)22-s + 0.253i·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $0.852 - 0.522i$ motivic weight = $$4$$ character : $\chi_{108} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 0.852 - 0.522i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.526598 + 0.148672i$$ $$L(\frac12)$$ $$\approx$$ $$0.526598 + 0.148672i$$ $$L(3)$$ 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 + (3.49 + 1.95i)T$$
3 $$1$$
good5 $$1 + 47.1T + 625T^{2}$$
7 $$1 + 67.4iT - 2.40e3T^{2}$$
11 $$1 - 69.3iT - 1.46e4T^{2}$$
13 $$1 - 3.28T + 2.85e4T^{2}$$
17 $$1 - 116.T + 8.35e4T^{2}$$
19 $$1 - 513. iT - 1.30e5T^{2}$$
23 $$1 - 134. iT - 2.79e5T^{2}$$
29 $$1 - 772.T + 7.07e5T^{2}$$
31 $$1 - 1.06e3iT - 9.23e5T^{2}$$
37 $$1 + 1.03e3T + 1.87e6T^{2}$$
41 $$1 - 2.27e3T + 2.82e6T^{2}$$
43 $$1 + 2.52e3iT - 3.41e6T^{2}$$
47 $$1 + 694. iT - 4.87e6T^{2}$$
53 $$1 + 1.16e3T + 7.89e6T^{2}$$
59 $$1 - 5.53e3iT - 1.21e7T^{2}$$
61 $$1 + 3.20e3T + 1.38e7T^{2}$$
67 $$1 - 5.36e3iT - 2.01e7T^{2}$$
71 $$1 + 166. iT - 2.54e7T^{2}$$
73 $$1 - 5.08e3T + 2.83e7T^{2}$$
79 $$1 + 724. iT - 3.89e7T^{2}$$
83 $$1 - 8.15e3iT - 4.74e7T^{2}$$
89 $$1 + 6.96e3T + 6.27e7T^{2}$$
97 $$1 + 278.T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}