# Properties

 Label 2-108-12.11-c3-0-15 Degree $2$ Conductor $108$ Sign $0.955 + 0.293i$ 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 + (−0.419 + 2.79i)2-s + (−7.64 − 2.34i)4-s − 1.49i·5-s − 26.1i·7-s + (9.78 − 20.4i)8-s + (4.18 + 0.627i)10-s + 56.3·11-s − 41.3·13-s + (73.2 + 10.9i)14-s + (52.9 + 35.9i)16-s − 51.0i·17-s − 79.0i·19-s + (−3.51 + 11.4i)20-s + (−23.6 + 157. i)22-s + 27.3·23-s + ⋯
 L(s)  = 1 + (−0.148 + 0.988i)2-s + (−0.955 − 0.293i)4-s − 0.133i·5-s − 1.41i·7-s + (0.432 − 0.901i)8-s + (0.132 + 0.0198i)10-s + 1.54·11-s − 0.881·13-s + (1.39 + 0.209i)14-s + (0.827 + 0.561i)16-s − 0.728i·17-s − 0.954i·19-s + (−0.0392 + 0.127i)20-s + (−0.229 + 1.52i)22-s + 0.248·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $0.955 + 0.293i$ 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: $$(2,\ 108,\ (\ :3/2),\ 0.955 + 0.293i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.21623 - 0.182582i$$ $$L(\frac12)$$ $$\approx$$ $$1.21623 - 0.182582i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.419 - 2.79i)T$$
3 $$1$$
good5 $$1 + 1.49iT - 125T^{2}$$
7 $$1 + 26.1iT - 343T^{2}$$
11 $$1 - 56.3T + 1.33e3T^{2}$$
13 $$1 + 41.3T + 2.19e3T^{2}$$
17 $$1 + 51.0iT - 4.91e3T^{2}$$
19 $$1 + 79.0iT - 6.85e3T^{2}$$
23 $$1 - 27.3T + 1.21e4T^{2}$$
29 $$1 - 134. iT - 2.43e4T^{2}$$
31 $$1 + 187. iT - 2.97e4T^{2}$$
37 $$1 + 196.T + 5.06e4T^{2}$$
41 $$1 + 298. iT - 6.89e4T^{2}$$
43 $$1 + 465. iT - 7.95e4T^{2}$$
47 $$1 + 373.T + 1.03e5T^{2}$$
53 $$1 - 620. iT - 1.48e5T^{2}$$
59 $$1 + 321.T + 2.05e5T^{2}$$
61 $$1 - 674.T + 2.26e5T^{2}$$
67 $$1 - 576. iT - 3.00e5T^{2}$$
71 $$1 - 223.T + 3.57e5T^{2}$$
73 $$1 - 70.1T + 3.89e5T^{2}$$
79 $$1 - 1.05e3iT - 4.93e5T^{2}$$
83 $$1 - 1.21e3T + 5.71e5T^{2}$$
89 $$1 - 1.34e3iT - 7.04e5T^{2}$$
97 $$1 + 576.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$