Properties

 Label 2-21-21.20-c21-0-30 Degree $2$ Conductor $21$ Sign $-0.138 - 0.990i$ Analytic cond. $58.6902$ Root an. cond. $7.66095$ Motivic weight $21$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 247. i·2-s + (−1.62e4 + 1.00e5i)3-s + 2.03e6·4-s + 3.48e7·5-s + (−2.49e7 − 4.00e6i)6-s + (7.14e8 − 2.19e8i)7-s + 1.02e9i·8-s + (−9.93e9 − 3.27e9i)9-s + 8.59e9i·10-s + 9.21e10i·11-s + (−3.30e10 + 2.05e11i)12-s + 1.67e11i·13-s + (5.41e10 + 1.76e11i)14-s + (−5.64e11 + 3.51e12i)15-s + 4.01e12·16-s − 7.92e12·17-s + ⋯
 L(s)  = 1 + 0.170i·2-s + (−0.158 + 0.987i)3-s + 0.970·4-s + 1.59·5-s + (−0.168 − 0.0270i)6-s + (0.955 − 0.293i)7-s + 0.336i·8-s + (−0.949 − 0.312i)9-s + 0.271i·10-s + 1.07i·11-s + (−0.153 + 0.958i)12-s + 0.337i·13-s + (0.0500 + 0.163i)14-s + (−0.252 + 1.57i)15-s + 0.913·16-s − 0.953·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $-0.138 - 0.990i$ Analytic conductor: $$58.6902$$ Root analytic conductor: $$7.66095$$ Motivic weight: $$21$$ Rational: no Arithmetic: yes Character: $\chi_{21} (20, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :21/2),\ -0.138 - 0.990i)$$

Particular Values

 $$L(11)$$ $$\approx$$ $$3.835170821$$ $$L(\frac12)$$ $$\approx$$ $$3.835170821$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.62e4 - 1.00e5i)T$$
7 $$1 + (-7.14e8 + 2.19e8i)T$$
good2 $$1 - 247. iT - 2.09e6T^{2}$$
5 $$1 - 3.48e7T + 4.76e14T^{2}$$
11 $$1 - 9.21e10iT - 7.40e21T^{2}$$
13 $$1 - 1.67e11iT - 2.47e23T^{2}$$
17 $$1 + 7.92e12T + 6.90e25T^{2}$$
19 $$1 - 3.32e13iT - 7.14e26T^{2}$$
23 $$1 + 3.78e14iT - 3.94e28T^{2}$$
29 $$1 - 2.17e15iT - 5.13e30T^{2}$$
31 $$1 - 3.73e15iT - 2.08e31T^{2}$$
37 $$1 + 1.35e16T + 8.55e32T^{2}$$
41 $$1 - 6.54e16T + 7.38e33T^{2}$$
43 $$1 - 4.14e15T + 2.00e34T^{2}$$
47 $$1 - 4.66e17T + 1.30e35T^{2}$$
53 $$1 + 1.25e18iT - 1.62e36T^{2}$$
59 $$1 + 5.49e17T + 1.54e37T^{2}$$
61 $$1 - 7.00e18iT - 3.10e37T^{2}$$
67 $$1 + 1.30e19T + 2.22e38T^{2}$$
71 $$1 + 3.38e19iT - 7.52e38T^{2}$$
73 $$1 - 5.69e19iT - 1.34e39T^{2}$$
79 $$1 + 5.19e19T + 7.08e39T^{2}$$
83 $$1 + 1.77e20T + 1.99e40T^{2}$$
89 $$1 - 4.40e20T + 8.65e40T^{2}$$
97 $$1 + 5.59e20iT - 5.27e41T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$