Properties

 Label 2-21-21.20-c21-0-37 Degree $2$ Conductor $21$ Sign $0.659 + 0.751i$ 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 + 1.02e5i·3-s + 2.09e6·4-s + (−5.61e8 + 4.92e8i)7-s − 1.04e10·9-s + 2.14e11i·12-s − 9.22e11i·13-s + 4.39e12·16-s − 3.99e13i·19-s + (−5.03e13 − 5.74e13i)21-s − 4.76e14·25-s − 1.06e15i·27-s + (−1.17e15 + 1.03e15i)28-s + 1.25e15i·31-s − 2.19e16·36-s + 5.77e16·37-s + ⋯
 L(s)  = 1 + 0.999i·3-s + 4-s + (−0.751 + 0.659i)7-s − 0.999·9-s + 0.999i·12-s − 1.85i·13-s + 16-s − 1.49i·19-s + (−0.659 − 0.751i)21-s − 0.999·25-s − 0.999i·27-s + (−0.751 + 0.659i)28-s + 0.275i·31-s − 0.999·36-s + 1.97·37-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.659 + 0.751i$ 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: $$(2,\ 21,\ (\ :21/2),\ 0.659 + 0.751i)$$

Particular Values

 $$L(11)$$ $$\approx$$ $$1.587701918$$ $$L(\frac12)$$ $$\approx$$ $$1.587701918$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 1.02e5iT$$
7 $$1 + (5.61e8 - 4.92e8i)T$$
good2 $$1 - 2.09e6T^{2}$$
5 $$1 + 4.76e14T^{2}$$
11 $$1 - 7.40e21T^{2}$$
13 $$1 + 9.22e11iT - 2.47e23T^{2}$$
17 $$1 + 6.90e25T^{2}$$
19 $$1 + 3.99e13iT - 7.14e26T^{2}$$
23 $$1 - 3.94e28T^{2}$$
29 $$1 - 5.13e30T^{2}$$
31 $$1 - 1.25e15iT - 2.08e31T^{2}$$
37 $$1 - 5.77e16T + 8.55e32T^{2}$$
41 $$1 + 7.38e33T^{2}$$
43 $$1 + 2.65e17T + 2.00e34T^{2}$$
47 $$1 + 1.30e35T^{2}$$
53 $$1 - 1.62e36T^{2}$$
59 $$1 + 1.54e37T^{2}$$
61 $$1 + 2.49e18iT - 3.10e37T^{2}$$
67 $$1 - 6.94e18T + 2.22e38T^{2}$$
71 $$1 - 7.52e38T^{2}$$
73 $$1 + 6.21e19iT - 1.34e39T^{2}$$
79 $$1 - 1.68e20T + 7.08e39T^{2}$$
83 $$1 + 1.99e40T^{2}$$
89 $$1 + 8.65e40T^{2}$$
97 $$1 + 9.10e20iT - 5.27e41T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$