Properties

 Label 2-6-3.2-c24-0-2 Degree $2$ Conductor $6$ Sign $0.454 - 0.890i$ Analytic cond. $21.8980$ Root an. cond. $4.67953$ Motivic weight $24$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2.89e3i·2-s + (4.73e5 + 2.41e5i)3-s − 8.38e6·4-s + 7.85e7i·5-s + (7.00e8 − 1.37e9i)6-s + 2.19e9·7-s + 2.42e10i·8-s + (1.65e11 + 2.28e11i)9-s + 2.27e11·10-s + 1.55e12i·11-s + (−3.97e12 − 2.02e12i)12-s − 3.72e13·13-s − 6.36e12i·14-s + (−1.89e13 + 3.71e13i)15-s + 7.03e13·16-s + 6.73e14i·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (0.890 + 0.454i)3-s − 0.500·4-s + 0.321i·5-s + (0.321 − 0.629i)6-s + 0.158·7-s + 0.353i·8-s + (0.586 + 0.810i)9-s + 0.227·10-s + 0.496i·11-s + (−0.445 − 0.227i)12-s − 1.59·13-s − 0.112i·14-s + (−0.146 + 0.286i)15-s + 0.250·16-s + 1.15i·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(25-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$6$$    =    $$2 \cdot 3$$ Sign: $0.454 - 0.890i$ Analytic conductor: $$21.8980$$ Root analytic conductor: $$4.67953$$ Motivic weight: $$24$$ Rational: no Arithmetic: yes Character: $\chi_{6} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 6,\ (\ :12),\ 0.454 - 0.890i)$$

Particular Values

 $$L(\frac{25}{2})$$ $$\approx$$ $$1.72435 + 1.05549i$$ $$L(\frac12)$$ $$\approx$$ $$1.72435 + 1.05549i$$ $$L(13)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2.89e3iT$$
3 $$1 + (-4.73e5 - 2.41e5i)T$$
good5 $$1 - 7.85e7iT - 5.96e16T^{2}$$
7 $$1 - 2.19e9T + 1.91e20T^{2}$$
11 $$1 - 1.55e12iT - 9.84e24T^{2}$$
13 $$1 + 3.72e13T + 5.42e26T^{2}$$
17 $$1 - 6.73e14iT - 3.39e29T^{2}$$
19 $$1 - 9.37e14T + 4.89e30T^{2}$$
23 $$1 - 3.13e16iT - 4.80e32T^{2}$$
29 $$1 - 2.21e17iT - 1.25e35T^{2}$$
31 $$1 - 6.76e17T + 6.20e35T^{2}$$
37 $$1 + 4.23e18T + 4.33e37T^{2}$$
41 $$1 + 2.43e19iT - 5.09e38T^{2}$$
43 $$1 - 2.27e19T + 1.59e39T^{2}$$
47 $$1 - 6.93e19iT - 1.35e40T^{2}$$
53 $$1 - 8.43e20iT - 2.41e41T^{2}$$
59 $$1 + 2.79e21iT - 3.16e42T^{2}$$
61 $$1 + 4.56e21T + 7.04e42T^{2}$$
67 $$1 + 1.20e21T + 6.69e43T^{2}$$
71 $$1 + 1.60e22iT - 2.69e44T^{2}$$
73 $$1 - 2.60e22T + 5.24e44T^{2}$$
79 $$1 - 8.00e22T + 3.49e45T^{2}$$
83 $$1 + 1.65e23iT - 1.14e46T^{2}$$
89 $$1 + 4.33e23iT - 6.10e46T^{2}$$
97 $$1 + 4.08e23T + 4.81e47T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$