Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2}$ Sign $-0.639 - 0.769i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (1.04 − 3.86i)2-s + (−7.79 + 4.49i)3-s + (−13.8 − 8.03i)4-s + (−5.89 + 10.2i)5-s + (9.23 + 34.7i)6-s + (−50.5 + 29.1i)7-s + (−45.4 + 45.0i)8-s + (40.6 − 70.0i)9-s + (33.2 + 33.3i)10-s + (−86.9 + 50.2i)11-s + (143. + 0.544i)12-s + (85.3 − 147. i)13-s + (60.1 + 225. i)14-s + (0.116 − 106. i)15-s + (126. + 222. i)16-s − 398.·17-s + ⋯
 L(s)  = 1 + (0.260 − 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.864 − 0.502i)4-s + (−0.235 + 0.408i)5-s + (0.256 + 0.966i)6-s + (−1.03 + 0.595i)7-s + (−0.709 + 0.704i)8-s + (0.501 − 0.864i)9-s + (0.332 + 0.333i)10-s + (−0.718 + 0.414i)11-s + (0.999 + 0.00378i)12-s + (0.504 − 0.874i)13-s + (0.306 + 1.15i)14-s + (0.000517 − 0.471i)15-s + (0.495 + 0.868i)16-s − 1.37·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$36$$    =    $$2^{2} \cdot 3^{2}$$ $$\varepsilon$$ = $-0.639 - 0.769i$ motivic weight = $$4$$ character : $\chi_{36} (31, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 36,\ (\ :2),\ -0.639 - 0.769i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.0423638 + 0.0902737i$$ $$L(\frac12)$$ $$\approx$$ $$0.0423638 + 0.0902737i$$ $$L(3)$$ 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 + (-1.04 + 3.86i)T$$
3 $$1 + (7.79 - 4.49i)T$$
good5 $$1 + (5.89 - 10.2i)T + (-312.5 - 541. i)T^{2}$$
7 $$1 + (50.5 - 29.1i)T + (1.20e3 - 2.07e3i)T^{2}$$
11 $$1 + (86.9 - 50.2i)T + (7.32e3 - 1.26e4i)T^{2}$$
13 $$1 + (-85.3 + 147. i)T + (-1.42e4 - 2.47e4i)T^{2}$$
17 $$1 + 398.T + 8.35e4T^{2}$$
19 $$1 + 404. iT - 1.30e5T^{2}$$
23 $$1 + (291. + 168. i)T + (1.39e5 + 2.42e5i)T^{2}$$
29 $$1 + (-327. - 567. i)T + (-3.53e5 + 6.12e5i)T^{2}$$
31 $$1 + (-550. - 317. i)T + (4.61e5 + 7.99e5i)T^{2}$$
37 $$1 + 1.59e3T + 1.87e6T^{2}$$
41 $$1 + (1.23e3 - 2.13e3i)T + (-1.41e6 - 2.44e6i)T^{2}$$
43 $$1 + (-1.93e3 + 1.11e3i)T + (1.70e6 - 2.96e6i)T^{2}$$
47 $$1 + (2.51e3 - 1.45e3i)T + (2.43e6 - 4.22e6i)T^{2}$$
53 $$1 - 1.29e3T + 7.89e6T^{2}$$
59 $$1 + (-1.00e3 - 578. i)T + (6.05e6 + 1.04e7i)T^{2}$$
61 $$1 + (2.96e3 + 5.12e3i)T + (-6.92e6 + 1.19e7i)T^{2}$$
67 $$1 + (3.08e3 + 1.78e3i)T + (1.00e7 + 1.74e7i)T^{2}$$
71 $$1 + 5.63e3iT - 2.54e7T^{2}$$
73 $$1 + 5.49e3T + 2.83e7T^{2}$$
79 $$1 + (-2.78e3 + 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2}$$
83 $$1 + (7.06e3 - 4.07e3i)T + (2.37e7 - 4.11e7i)T^{2}$$
89 $$1 - 910.T + 6.27e7T^{2}$$
97 $$1 + (-8.80e3 - 1.52e4i)T + (-4.42e7 + 7.66e7i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}