Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.531 + 0.847i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−5.48 − 1.40i)2-s + (28.0 + 15.3i)4-s + (77.8 − 44.9i)5-s + (−124. − 71.6i)7-s + (−132. − 123. i)8-s + (−489. + 137. i)10-s + (278. − 481. i)11-s + (179. + 311. i)13-s + (579. + 566. i)14-s + (551. + 863. i)16-s + 1.07e3i·17-s − 348. i·19-s + (2.87e3 − 64.1i)20-s + (−2.20e3 + 2.25e3i)22-s + (−1.46e3 − 2.54e3i)23-s + ⋯
 L(s)  = 1 + (−0.968 − 0.248i)2-s + (0.876 + 0.480i)4-s + (1.39 − 0.803i)5-s + (−0.956 − 0.552i)7-s + (−0.730 − 0.683i)8-s + (−1.54 + 0.433i)10-s + (0.693 − 1.20i)11-s + (0.295 + 0.511i)13-s + (0.789 + 0.772i)14-s + (0.538 + 0.842i)16-s + 0.903i·17-s − 0.221i·19-s + (1.60 − 0.0358i)20-s + (−0.969 + 0.991i)22-s + (−0.579 − 1.00i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.531 + 0.847i$ motivic weight = $$5$$ character : $\chi_{108} (35, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ -0.531 + 0.847i)$$ $$L(3)$$ $$\approx$$ $$0.561719 - 1.01568i$$ $$L(\frac12)$$ $$\approx$$ $$0.561719 - 1.01568i$$ $$L(\frac{7}{2})$$ 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 + (5.48 + 1.40i)T$$
3 $$1$$
good5 $$1 + (-77.8 + 44.9i)T + (1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (124. + 71.6i)T + (8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (-278. + 481. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (-179. - 311. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 - 1.07e3iT - 1.41e6T^{2}$$
19 $$1 + 348. iT - 2.47e6T^{2}$$
23 $$1 + (1.46e3 + 2.54e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (3.33e3 + 1.92e3i)T + (1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (2.34e3 - 1.35e3i)T + (1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + 1.99e3T + 6.93e7T^{2}$$
41 $$1 + (-1.34e4 + 7.75e3i)T + (5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (1.24e4 + 7.20e3i)T + (7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (-1.23e3 + 2.13e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + 1.33e4iT - 4.18e8T^{2}$$
59 $$1 + (1.98e4 + 3.44e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (211. - 366. i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-1.40e4 + 8.09e3i)T + (6.75e8 - 1.16e9i)T^{2}$$
71 $$1 - 6.90e3T + 1.80e9T^{2}$$
73 $$1 + 1.10e4T + 2.07e9T^{2}$$
79 $$1 + (-6.43e4 - 3.71e4i)T + (1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (4.85e4 - 8.41e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + 7.44e4iT - 5.58e9T^{2}$$
97 $$1 + (-4.79e4 + 8.30e4i)T + (-4.29e9 - 7.43e9i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}