# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2}$ Sign $0.481 - 0.876i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.89 + 0.903i)2-s + (−8.98 + 0.573i)3-s + (14.3 + 7.03i)4-s + (19.5 + 33.8i)5-s + (−35.5 − 5.87i)6-s + (10.5 + 6.10i)7-s + (49.6 + 40.4i)8-s + (80.3 − 10.3i)9-s + (45.5 + 149. i)10-s + (−96.1 − 55.5i)11-s + (−133. − 54.9i)12-s + (−104. − 180. i)13-s + (35.6 + 33.3i)14-s + (−194. − 292. i)15-s + (156. + 202. i)16-s + 93.3·17-s + ⋯
 L(s)  = 1 + (0.974 + 0.225i)2-s + (−0.997 + 0.0637i)3-s + (0.898 + 0.439i)4-s + (0.781 + 1.35i)5-s + (−0.986 − 0.163i)6-s + (0.215 + 0.124i)7-s + (0.775 + 0.631i)8-s + (0.991 − 0.127i)9-s + (0.455 + 1.49i)10-s + (−0.794 − 0.458i)11-s + (−0.924 − 0.381i)12-s + (−0.618 − 1.07i)13-s + (0.182 + 0.170i)14-s + (−0.866 − 1.30i)15-s + (0.613 + 0.790i)16-s + 0.323·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$36$$    =    $$2^{2} \cdot 3^{2}$$ $$\varepsilon$$ = $0.481 - 0.876i$ motivic weight = $$4$$ character : $\chi_{36} (7, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 36,\ (\ :2),\ 0.481 - 0.876i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.78628 + 1.05675i$$ $$L(\frac12)$$ $$\approx$$ $$1.78628 + 1.05675i$$ $$L(3)$$ 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 + (-3.89 - 0.903i)T$$
3 $$1 + (8.98 - 0.573i)T$$
good5 $$1 + (-19.5 - 33.8i)T + (-312.5 + 541. i)T^{2}$$
7 $$1 + (-10.5 - 6.10i)T + (1.20e3 + 2.07e3i)T^{2}$$
11 $$1 + (96.1 + 55.5i)T + (7.32e3 + 1.26e4i)T^{2}$$
13 $$1 + (104. + 180. i)T + (-1.42e4 + 2.47e4i)T^{2}$$
17 $$1 - 93.3T + 8.35e4T^{2}$$
19 $$1 + 26.8iT - 1.30e5T^{2}$$
23 $$1 + (-757. + 437. i)T + (1.39e5 - 2.42e5i)T^{2}$$
29 $$1 + (-650. + 1.12e3i)T + (-3.53e5 - 6.12e5i)T^{2}$$
31 $$1 + (593. - 342. i)T + (4.61e5 - 7.99e5i)T^{2}$$
37 $$1 + 1.76e3T + 1.87e6T^{2}$$
41 $$1 + (39.0 + 67.6i)T + (-1.41e6 + 2.44e6i)T^{2}$$
43 $$1 + (-1.40e3 - 811. i)T + (1.70e6 + 2.96e6i)T^{2}$$
47 $$1 + (-1.99e3 - 1.15e3i)T + (2.43e6 + 4.22e6i)T^{2}$$
53 $$1 + 1.31e3T + 7.89e6T^{2}$$
59 $$1 + (4.81e3 - 2.78e3i)T + (6.05e6 - 1.04e7i)T^{2}$$
61 $$1 + (1.09e3 - 1.88e3i)T + (-6.92e6 - 1.19e7i)T^{2}$$
67 $$1 + (-213. + 123. i)T + (1.00e7 - 1.74e7i)T^{2}$$
71 $$1 + 4.60e3iT - 2.54e7T^{2}$$
73 $$1 - 2.56e3T + 2.83e7T^{2}$$
79 $$1 + (4.48e3 + 2.59e3i)T + (1.94e7 + 3.37e7i)T^{2}$$
83 $$1 + (-1.62e3 - 936. i)T + (2.37e7 + 4.11e7i)T^{2}$$
89 $$1 + 1.16e3T + 6.27e7T^{2}$$
97 $$1 + (-2.86e3 + 4.97e3i)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}