# Properties

 Label 2-7-7.6-c6-0-2 Degree $2$ Conductor $7$ Sign $-0.387 + 0.921i$ Analytic cond. $1.61037$ Root an. cond. $1.26900$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 8·2-s − 45.1i·3-s + 45.1i·5-s + 361. i·6-s + (133 − 316. i)7-s + 512·8-s − 1.31e3·9-s − 361. i·10-s + 874·11-s − 2.21e3i·13-s + (−1.06e3 + 2.52e3i)14-s + 2.03e3·15-s − 4.09e3·16-s + 5.96e3i·17-s + 1.04e4·18-s − 3.11e3i·19-s + ⋯
 L(s)  = 1 − 2-s − 1.67i·3-s + 0.361i·5-s + 1.67i·6-s + (0.387 − 0.921i)7-s + 8-s − 1.79·9-s − 0.361i·10-s + 0.656·11-s − 1.00i·13-s + (−0.387 + 0.921i)14-s + 0.604·15-s − 16-s + 1.21i·17-s + 1.79·18-s − 0.454i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$7$$ Sign: $-0.387 + 0.921i$ Analytic conductor: $$1.61037$$ Root analytic conductor: $$1.26900$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{7} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 7,\ (\ :3),\ -0.387 + 0.921i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.369522 - 0.556333i$$ $$L(\frac12)$$ $$\approx$$ $$0.369522 - 0.556333i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-133 + 316. i)T$$
good2 $$1 + 8T + 64T^{2}$$
3 $$1 + 45.1iT - 729T^{2}$$
5 $$1 - 45.1iT - 1.56e4T^{2}$$
11 $$1 - 874T + 1.77e6T^{2}$$
13 $$1 + 2.21e3iT - 4.82e6T^{2}$$
17 $$1 - 5.96e3iT - 2.41e7T^{2}$$
19 $$1 + 3.11e3iT - 4.70e7T^{2}$$
23 $$1 - 4.73e3T + 1.48e8T^{2}$$
29 $$1 - 1.11e4T + 5.94e8T^{2}$$
31 $$1 - 2.74e4iT - 8.87e8T^{2}$$
37 $$1 - 3.00e3T + 2.56e9T^{2}$$
41 $$1 + 5.75e4iT - 4.75e9T^{2}$$
43 $$1 - 3.14e4T + 6.32e9T^{2}$$
47 $$1 - 7.24e4iT - 1.07e10T^{2}$$
53 $$1 + 7.64e4T + 2.21e10T^{2}$$
59 $$1 + 1.13e5iT - 4.21e10T^{2}$$
61 $$1 - 2.75e5iT - 5.15e10T^{2}$$
67 $$1 - 4.95e5T + 9.04e10T^{2}$$
71 $$1 + 1.84e5T + 1.28e11T^{2}$$
73 $$1 - 6.09e4iT - 1.51e11T^{2}$$
79 $$1 + 5.34e5T + 2.43e11T^{2}$$
83 $$1 - 7.14e5iT - 3.26e11T^{2}$$
89 $$1 - 6.29e5iT - 4.96e11T^{2}$$
97 $$1 + 8.14e5iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$