# Properties

 Label 2-108-12.11-c3-0-16 Degree $2$ Conductor $108$ Sign $-0.850 + 0.526i$ Analytic cond. $6.37220$ Root an. cond. $2.52432$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.72 + 0.773i)2-s + (6.80 − 4.21i)4-s − 20.8i·5-s + 13.9i·7-s + (−15.2 + 16.7i)8-s + (16.1 + 56.7i)10-s − 34.5·11-s − 31.3·13-s + (−10.7 − 37.8i)14-s + (28.5 − 57.2i)16-s + 34.4i·17-s − 120. i·19-s + (−87.7 − 141. i)20-s + (93.8 − 26.7i)22-s − 137.·23-s + ⋯
 L(s)  = 1 + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s − 1.86i·5-s + 0.750i·7-s + (−0.673 + 0.738i)8-s + (0.510 + 1.79i)10-s − 0.945·11-s − 0.668·13-s + (−0.205 − 0.722i)14-s + (0.445 − 0.895i)16-s + 0.491i·17-s − 1.45i·19-s + (−0.981 − 1.58i)20-s + (0.909 − 0.258i)22-s − 1.24·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $-0.850 + 0.526i$ Analytic conductor: $$6.37220$$ Root analytic conductor: $$2.52432$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{108} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 108,\ (\ :3/2),\ -0.850 + 0.526i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.120282 - 0.422795i$$ $$L(\frac12)$$ $$\approx$$ $$0.120282 - 0.422795i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (2.72 - 0.773i)T$$
3 $$1$$
good5 $$1 + 20.8iT - 125T^{2}$$
7 $$1 - 13.9iT - 343T^{2}$$
11 $$1 + 34.5T + 1.33e3T^{2}$$
13 $$1 + 31.3T + 2.19e3T^{2}$$
17 $$1 - 34.4iT - 4.91e3T^{2}$$
19 $$1 + 120. iT - 6.85e3T^{2}$$
23 $$1 + 137.T + 1.21e4T^{2}$$
29 $$1 + 93.1iT - 2.43e4T^{2}$$
31 $$1 - 111. iT - 2.97e4T^{2}$$
37 $$1 + 146.T + 5.06e4T^{2}$$
41 $$1 + 8.44iT - 6.89e4T^{2}$$
43 $$1 + 427. iT - 7.95e4T^{2}$$
47 $$1 - 318.T + 1.03e5T^{2}$$
53 $$1 - 291. iT - 1.48e5T^{2}$$
59 $$1 + 364.T + 2.05e5T^{2}$$
61 $$1 + 289.T + 2.26e5T^{2}$$
67 $$1 + 305. iT - 3.00e5T^{2}$$
71 $$1 - 102.T + 3.57e5T^{2}$$
73 $$1 - 442.T + 3.89e5T^{2}$$
79 $$1 + 245. iT - 4.93e5T^{2}$$
83 $$1 - 478.T + 5.71e5T^{2}$$
89 $$1 + 1.41e3iT - 7.04e5T^{2}$$
97 $$1 - 1.15e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$