# Properties

 Degree $2$ Conductor $108$ Sign $-0.633 + 0.774i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (6.98 − 5.67i)3-s + (−8.75 − 24.0i)5-s + (−0.632 − 3.58i)7-s + (16.4 − 79.3i)9-s + (−22.6 + 62.1i)11-s + (−236. − 198. i)13-s + (−197. − 118. i)15-s + (205. + 118. i)17-s + (25.2 + 43.7i)19-s + (−24.7 − 21.4i)21-s + (−330. − 58.2i)23-s + (−23.6 + 19.8i)25-s + (−335. − 647. i)27-s + (−402. − 480. i)29-s + (111. − 634. i)31-s + ⋯
 L(s)  = 1 + (0.775 − 0.631i)3-s + (−0.350 − 0.962i)5-s + (−0.0129 − 0.0731i)7-s + (0.203 − 0.979i)9-s + (−0.187 + 0.514i)11-s + (−1.39 − 1.17i)13-s + (−0.879 − 0.525i)15-s + (0.710 + 0.410i)17-s + (0.0699 + 0.121i)19-s + (−0.0561 − 0.0486i)21-s + (−0.624 − 0.110i)23-s + (−0.0378 + 0.0317i)25-s + (−0.460 − 0.887i)27-s + (−0.479 − 0.570i)29-s + (0.116 − 0.660i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $-0.633 + 0.774i$ Motivic weight: $$4$$ Character: $\chi_{108} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 108,\ (\ :2),\ -0.633 + 0.774i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.714858 - 1.50795i$$ $$L(\frac12)$$ $$\approx$$ $$0.714858 - 1.50795i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-6.98 + 5.67i)T$$
good5 $$1 + (8.75 + 24.0i)T + (-478. + 401. i)T^{2}$$
7 $$1 + (0.632 + 3.58i)T + (-2.25e3 + 821. i)T^{2}$$
11 $$1 + (22.6 - 62.1i)T + (-1.12e4 - 9.41e3i)T^{2}$$
13 $$1 + (236. + 198. i)T + (4.95e3 + 2.81e4i)T^{2}$$
17 $$1 + (-205. - 118. i)T + (4.17e4 + 7.23e4i)T^{2}$$
19 $$1 + (-25.2 - 43.7i)T + (-6.51e4 + 1.12e5i)T^{2}$$
23 $$1 + (330. + 58.2i)T + (2.62e5 + 9.57e4i)T^{2}$$
29 $$1 + (402. + 480. i)T + (-1.22e5 + 6.96e5i)T^{2}$$
31 $$1 + (-111. + 634. i)T + (-8.67e5 - 3.15e5i)T^{2}$$
37 $$1 + (-978. + 1.69e3i)T + (-9.37e5 - 1.62e6i)T^{2}$$
41 $$1 + (724. - 862. i)T + (-4.90e5 - 2.78e6i)T^{2}$$
43 $$1 + (-296. - 107. i)T + (2.61e6 + 2.19e6i)T^{2}$$
47 $$1 + (-3.61e3 + 636. i)T + (4.58e6 - 1.66e6i)T^{2}$$
53 $$1 - 3.92e3iT - 7.89e6T^{2}$$
59 $$1 + (-662. - 1.82e3i)T + (-9.28e6 + 7.78e6i)T^{2}$$
61 $$1 + (-633. - 3.59e3i)T + (-1.30e7 + 4.73e6i)T^{2}$$
67 $$1 + (1.25e3 + 1.05e3i)T + (3.49e6 + 1.98e7i)T^{2}$$
71 $$1 + (-6.35e3 - 3.66e3i)T + (1.27e7 + 2.20e7i)T^{2}$$
73 $$1 + (832. + 1.44e3i)T + (-1.41e7 + 2.45e7i)T^{2}$$
79 $$1 + (-8.24e3 + 6.91e3i)T + (6.76e6 - 3.83e7i)T^{2}$$
83 $$1 + (-4.03e3 - 4.81e3i)T + (-8.24e6 + 4.67e7i)T^{2}$$
89 $$1 + (-4.49e3 + 2.59e3i)T + (3.13e7 - 5.43e7i)T^{2}$$
97 $$1 + (3.95e3 + 1.43e3i)T + (6.78e7 + 5.69e7i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$