# Properties

 Label 2-21-21.11-c6-0-7 Degree $2$ Conductor $21$ Sign $0.859 - 0.510i$ Analytic cond. $4.83113$ Root an. cond. $2.19798$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.07 + 0.621i)2-s + (26.6 − 4.11i)3-s + (−31.2 + 54.0i)4-s + (93.7 − 54.1i)5-s + (−26.1 + 21.0i)6-s + (147. + 309. i)7-s − 157. i·8-s + (695. − 219. i)9-s + (−67.2 + 116. i)10-s + (1.89e3 + 1.09e3i)11-s + (−610. + 1.57e3i)12-s − 1.13e3·13-s + (−351. − 241. i)14-s + (2.27e3 − 1.83e3i)15-s + (−1.90e3 − 3.29e3i)16-s + (−6.94e3 − 4.01e3i)17-s + ⋯
 L(s)  = 1 + (−0.134 + 0.0776i)2-s + (0.988 − 0.152i)3-s + (−0.487 + 0.845i)4-s + (0.749 − 0.433i)5-s + (−0.121 + 0.0972i)6-s + (0.431 + 0.902i)7-s − 0.306i·8-s + (0.953 − 0.301i)9-s + (−0.0672 + 0.116i)10-s + (1.42 + 0.822i)11-s + (−0.353 + 0.909i)12-s − 0.518·13-s + (−0.128 − 0.0878i)14-s + (0.675 − 0.542i)15-s + (−0.464 − 0.803i)16-s + (−1.41 − 0.816i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$21$$    =    $$3 \cdot 7$$ Sign: $0.859 - 0.510i$ Analytic conductor: $$4.83113$$ Root analytic conductor: $$2.19798$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{21} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 21,\ (\ :3),\ 0.859 - 0.510i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.91597 + 0.525781i$$ $$L(\frac12)$$ $$\approx$$ $$1.91597 + 0.525781i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-26.6 + 4.11i)T$$
7 $$1 + (-147. - 309. i)T$$
good2 $$1 + (1.07 - 0.621i)T + (32 - 55.4i)T^{2}$$
5 $$1 + (-93.7 + 54.1i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (-1.89e3 - 1.09e3i)T + (8.85e5 + 1.53e6i)T^{2}$$
13 $$1 + 1.13e3T + 4.82e6T^{2}$$
17 $$1 + (6.94e3 + 4.01e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (3.23e3 + 5.60e3i)T + (-2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (-1.11e3 + 642. i)T + (7.40e7 - 1.28e8i)T^{2}$$
29 $$1 + 1.42e4iT - 5.94e8T^{2}$$
31 $$1 + (8.12e3 - 1.40e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (1.66e4 + 2.88e4i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 9.86e4iT - 4.75e9T^{2}$$
43 $$1 - 6.21e4T + 6.32e9T^{2}$$
47 $$1 + (5.03e4 - 2.90e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (-1.57e5 - 9.07e4i)T + (1.10e10 + 1.91e10i)T^{2}$$
59 $$1 + (1.27e5 + 7.36e4i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (-6.34e3 - 1.09e4i)T + (-2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (1.88e5 - 3.26e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + 1.18e5iT - 1.28e11T^{2}$$
73 $$1 + (-3.41e3 + 5.91e3i)T + (-7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (1.83e5 + 3.18e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + 3.36e5iT - 3.26e11T^{2}$$
89 $$1 + (2.58e5 - 1.49e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 + 5.39e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$