Properties

 Label 2-39-39.8-c3-0-4 Degree $2$ Conductor $39$ Sign $0.907 - 0.420i$ Analytic cond. $2.30107$ Root an. cond. $1.51692$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 5.19·3-s + 8i·4-s + (5.58 − 5.58i)7-s + 27·9-s + 41.5i·12-s + (−31.1 − 35i)13-s − 64·16-s + (−105. − 105. i)19-s + (29.0 − 29.0i)21-s + 125i·25-s + 140.·27-s + (44.7 + 44.7i)28-s + (76.0 + 76.0i)31-s + 216i·36-s + (273. − 273. i)37-s + ⋯
 L(s)  = 1 + 1.00·3-s + i·4-s + (0.301 − 0.301i)7-s + 9-s + 0.999i·12-s + (−0.665 − 0.746i)13-s − 16-s + (−1.27 − 1.27i)19-s + (0.301 − 0.301i)21-s + i·25-s + 1.00·27-s + (0.301 + 0.301i)28-s + (0.440 + 0.440i)31-s + i·36-s + (1.21 − 1.21i)37-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$39$$    =    $$3 \cdot 13$$ Sign: $0.907 - 0.420i$ Analytic conductor: $$2.30107$$ Root analytic conductor: $$1.51692$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{39} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 39,\ (\ :3/2),\ 0.907 - 0.420i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.63622 + 0.360475i$$ $$L(\frac12)$$ $$\approx$$ $$1.63622 + 0.360475i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 5.19T$$
13 $$1 + (31.1 + 35i)T$$
good2 $$1 - 8iT^{2}$$
5 $$1 - 125iT^{2}$$
7 $$1 + (-5.58 + 5.58i)T - 343iT^{2}$$
11 $$1 + 1.33e3iT^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 + (105. + 105. i)T + 6.85e3iT^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + (-76.0 - 76.0i)T + 2.97e4iT^{2}$$
37 $$1 + (-273. + 273. i)T - 5.06e4iT^{2}$$
41 $$1 - 6.89e4iT^{2}$$
43 $$1 - 218. iT - 7.95e4T^{2}$$
47 $$1 + 1.03e5iT^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 + 2.05e5iT^{2}$$
61 $$1 + 935.T + 2.26e5T^{2}$$
67 $$1 + (-767. - 767. i)T + 3.00e5iT^{2}$$
71 $$1 - 3.57e5iT^{2}$$
73 $$1 + (782. - 782. i)T - 3.89e5iT^{2}$$
79 $$1 - 1.09e3T + 4.93e5T^{2}$$
83 $$1 - 5.71e5iT^{2}$$
89 $$1 + 7.04e5iT^{2}$$
97 $$1 + (1.35e3 + 1.35e3i)T + 9.12e5iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$