# Properties

 Degree $2$ Conductor $36$ Sign $-0.131 + 0.991i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−5.35 − 1.83i)2-s + (6.54 − 14.1i)3-s + (25.2 + 19.6i)4-s + (36.0 − 20.8i)5-s + (−61.0 + 63.6i)6-s + (117. + 67.7i)7-s + (−99.1 − 151. i)8-s + (−157. − 185. i)9-s + (−231. + 45.2i)10-s + (144. − 249. i)11-s + (443. − 228. i)12-s + (−399. − 692. i)13-s + (−503. − 577. i)14-s + (−58.3 − 646. i)15-s + (252. + 992. i)16-s − 1.42e3i·17-s + ⋯
 L(s)  = 1 + (−0.945 − 0.324i)2-s + (0.420 − 0.907i)3-s + (0.789 + 0.613i)4-s + (0.645 − 0.372i)5-s + (−0.691 + 0.722i)6-s + (0.904 + 0.522i)7-s + (−0.547 − 0.836i)8-s + (−0.647 − 0.762i)9-s + (−0.730 + 0.143i)10-s + (0.359 − 0.622i)11-s + (0.888 − 0.458i)12-s + (−0.655 − 1.13i)13-s + (−0.686 − 0.787i)14-s + (−0.0670 − 0.741i)15-s + (0.246 + 0.969i)16-s − 1.19i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$36$$    =    $$2^{2} \cdot 3^{2}$$ Sign: $-0.131 + 0.991i$ Motivic weight: $$5$$ Character: $\chi_{36} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 36,\ (\ :5/2),\ -0.131 + 0.991i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.876375 - 1.00001i$$ $$L(\frac12)$$ $$\approx$$ $$0.876375 - 1.00001i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (5.35 + 1.83i)T$$
3 $$1 + (-6.54 + 14.1i)T$$
good5 $$1 + (-36.0 + 20.8i)T + (1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (-117. - 67.7i)T + (8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (-144. + 249. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (399. + 692. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + 1.42e3iT - 1.41e6T^{2}$$
19 $$1 - 2.19e3iT - 2.47e6T^{2}$$
23 $$1 + (276. + 478. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (-2.03e3 - 1.17e3i)T + (1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-8.14e3 + 4.70e3i)T + (1.43e7 - 2.47e7i)T^{2}$$
37 $$1 - 2.86e3T + 6.93e7T^{2}$$
41 $$1 + (2.49e3 - 1.44e3i)T + (5.79e7 - 1.00e8i)T^{2}$$
43 $$1 + (-5.63e3 - 3.25e3i)T + (7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (3.57e3 - 6.19e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 - 1.98e4iT - 4.18e8T^{2}$$
59 $$1 + (-2.46e4 - 4.26e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (1.93e4 - 3.34e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-3.70e4 + 2.13e4i)T + (6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 4.94e4T + 1.80e9T^{2}$$
73 $$1 - 5.26e4T + 2.07e9T^{2}$$
79 $$1 + (1.30e3 + 751. i)T + (1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (4.53e4 - 7.85e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 - 1.72e4iT - 5.58e9T^{2}$$
97 $$1 + (-4.25e4 + 7.37e4i)T + (-4.29e9 - 7.43e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$