Properties

 Label 2-108-9.7-c3-0-0 Degree $2$ Conductor $108$ Sign $-0.839 - 0.543i$ 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 + (−6.92 − 11.9i)5-s + (−15.3 + 26.5i)7-s + (−21.9 + 38.0i)11-s + (6.11 + 10.5i)13-s − 76.0·17-s − 44.1·19-s + (−39.3 − 68.0i)23-s + (−33.3 + 57.7i)25-s + (−46.3 + 80.3i)29-s + (71.5 + 123. i)31-s + 425.·35-s − 32.4·37-s + (−167. − 290. i)41-s + (249. − 431. i)43-s + (−140. + 244. i)47-s + ⋯
 L(s)  = 1 + (−0.619 − 1.07i)5-s + (−0.829 + 1.43i)7-s + (−0.601 + 1.04i)11-s + (0.130 + 0.226i)13-s − 1.08·17-s − 0.533·19-s + (−0.356 − 0.617i)23-s + (−0.266 + 0.461i)25-s + (−0.297 + 0.514i)29-s + (0.414 + 0.717i)31-s + 2.05·35-s − 0.144·37-s + (−0.639 − 1.10i)41-s + (0.883 − 1.53i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $-0.839 - 0.543i$ Analytic conductor: $$6.37220$$ Root analytic conductor: $$2.52432$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{108} (73, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 108,\ (\ :3/2),\ -0.839 - 0.543i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.101381 + 0.342747i$$ $$L(\frac12)$$ $$\approx$$ $$0.101381 + 0.342747i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + (6.92 + 11.9i)T + (-62.5 + 108. i)T^{2}$$
7 $$1 + (15.3 - 26.5i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (21.9 - 38.0i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-6.11 - 10.5i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 + 76.0T + 4.91e3T^{2}$$
19 $$1 + 44.1T + 6.85e3T^{2}$$
23 $$1 + (39.3 + 68.0i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (46.3 - 80.3i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-71.5 - 123. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + 32.4T + 5.06e4T^{2}$$
41 $$1 + (167. + 290. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-249. + 431. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (140. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 628.T + 1.48e5T^{2}$$
59 $$1 + (-252. - 437. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (185. - 322. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-81.3 - 140. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 433.T + 3.57e5T^{2}$$
73 $$1 + 629.T + 3.89e5T^{2}$$
79 $$1 + (86.3 - 149. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (87.4 - 151. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 + 336.T + 7.04e5T^{2}$$
97 $$1 + (42.1 - 73.0i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$