Properties

 Label 2-39-13.9-c3-0-0 Degree $2$ Conductor $39$ Sign $-0.999 - 0.0256i$ 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 + (−1.5 + 2.59i)2-s + (−1.5 + 2.59i)3-s + (−0.5 − 0.866i)4-s − 9·5-s + (−4.5 − 7.79i)6-s + (−1 − 1.73i)7-s − 21·8-s + (−4.5 − 7.79i)9-s + (13.5 − 23.3i)10-s + (−15 + 25.9i)11-s + 3.00·12-s + (32.5 + 33.7i)13-s + 6·14-s + (13.5 − 23.3i)15-s + (35.5 − 61.4i)16-s + (55.5 + 96.1i)17-s + ⋯
 L(s)  = 1 + (−0.530 + 0.918i)2-s + (−0.288 + 0.499i)3-s + (−0.0625 − 0.108i)4-s − 0.804·5-s + (−0.306 − 0.530i)6-s + (−0.0539 − 0.0935i)7-s − 0.928·8-s + (−0.166 − 0.288i)9-s + (0.426 − 0.739i)10-s + (−0.411 + 0.712i)11-s + 0.0721·12-s + (0.693 + 0.720i)13-s + 0.114·14-s + (0.232 − 0.402i)15-s + (0.554 − 0.960i)16-s + (0.791 + 1.37i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$39$$    =    $$3 \cdot 13$$ Sign: $-0.999 - 0.0256i$ Analytic conductor: $$2.30107$$ Root analytic conductor: $$1.51692$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{39} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 39,\ (\ :3/2),\ -0.999 - 0.0256i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.00810706 + 0.632160i$$ $$L(\frac12)$$ $$\approx$$ $$0.00810706 + 0.632160i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.5 - 2.59i)T$$
13 $$1 + (-32.5 - 33.7i)T$$
good2 $$1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + 9T + 125T^{2}$$
7 $$1 + (1 + 1.73i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2}$$
17 $$1 + (-55.5 - 96.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-23 - 39.8i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-3 + 5.19i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-52.5 + 90.9i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + 100T + 2.97e4T^{2}$$
37 $$1 + (8.5 - 14.7i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (-115.5 + 200. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-257 - 445. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + 162T + 1.03e5T^{2}$$
53 $$1 - 639T + 1.48e5T^{2}$$
59 $$1 + (300 + 519. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (116.5 + 201. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (463 - 801. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + (-465 - 805. i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 + 253T + 3.89e5T^{2}$$
79 $$1 + 1.32e3T + 4.93e5T^{2}$$
83 $$1 - 810T + 5.71e5T^{2}$$
89 $$1 + (249 - 431. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (679 + 1.17e3i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$