# Properties

 Degree $2$ Conductor $108$ Sign $-0.0796 - 0.996i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.63 + 1.03i)2-s + (5.16 − 0.555i)3-s + (5.86 − 5.43i)4-s + (6.86 + 18.8i)5-s + (−13.0 + 6.79i)6-s + (−18.0 + 3.17i)7-s + (−9.82 + 20.3i)8-s + (26.3 − 5.73i)9-s + (−37.5 − 42.5i)10-s + (−8.64 − 3.14i)11-s + (27.2 − 31.3i)12-s + (23.2 + 19.4i)13-s + (44.1 − 26.9i)14-s + (45.9 + 93.6i)15-s + (4.82 − 63.8i)16-s + (26.8 + 15.4i)17-s + ⋯
 L(s)  = 1 + (−0.930 + 0.365i)2-s + (0.994 − 0.106i)3-s + (0.733 − 0.679i)4-s + (0.614 + 1.68i)5-s + (−0.886 + 0.462i)6-s + (−0.972 + 0.171i)7-s + (−0.434 + 0.900i)8-s + (0.977 − 0.212i)9-s + (−1.18 − 1.34i)10-s + (−0.237 − 0.0862i)11-s + (0.656 − 0.754i)12-s + (0.495 + 0.415i)13-s + (0.842 − 0.514i)14-s + (0.790 + 1.61i)15-s + (0.0754 − 0.997i)16-s + (0.382 + 0.221i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $-0.0796 - 0.996i$ Motivic weight: $$3$$ Character: $\chi_{108} (59, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 108,\ (\ :3/2),\ -0.0796 - 0.996i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.950910 + 1.02996i$$ $$L(\frac12)$$ $$\approx$$ $$0.950910 + 1.02996i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (2.63 - 1.03i)T$$
3 $$1 + (-5.16 + 0.555i)T$$
good5 $$1 + (-6.86 - 18.8i)T + (-95.7 + 80.3i)T^{2}$$
7 $$1 + (18.0 - 3.17i)T + (322. - 117. i)T^{2}$$
11 $$1 + (8.64 + 3.14i)T + (1.01e3 + 855. i)T^{2}$$
13 $$1 + (-23.2 - 19.4i)T + (381. + 2.16e3i)T^{2}$$
17 $$1 + (-26.8 - 15.4i)T + (2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (113. - 65.3i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-9.01 + 51.1i)T + (-1.14e4 - 4.16e3i)T^{2}$$
29 $$1 + (-166. - 198. i)T + (-4.23e3 + 2.40e4i)T^{2}$$
31 $$1 + (-176. - 31.0i)T + (2.79e4 + 1.01e4i)T^{2}$$
37 $$1 + (-134. + 232. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (36.5 - 43.5i)T + (-1.19e4 - 6.78e4i)T^{2}$$
43 $$1 + (-120. + 331. i)T + (-6.09e4 - 5.11e4i)T^{2}$$
47 $$1 + (-41.3 - 234. i)T + (-9.75e4 + 3.55e4i)T^{2}$$
53 $$1 + 424. iT - 1.48e5T^{2}$$
59 $$1 + (79.6 - 28.9i)T + (1.57e5 - 1.32e5i)T^{2}$$
61 $$1 + (143. + 813. i)T + (-2.13e5 + 7.76e4i)T^{2}$$
67 $$1 + (-328. + 390. i)T + (-5.22e4 - 2.96e5i)T^{2}$$
71 $$1 + (-37.9 + 65.6i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 + (340. + 589. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-756. - 901. i)T + (-8.56e4 + 4.85e5i)T^{2}$$
83 $$1 + (-163. + 136. i)T + (9.92e4 - 5.63e5i)T^{2}$$
89 $$1 + (-719. + 415. i)T + (3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (-571. - 207. i)T + (6.99e5 + 5.86e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$