# Properties

 Degree $2$ Conductor $63$ Sign $-0.994 + 0.102i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.30 + 2.26i)2-s + (−3.13 + 4.14i)3-s + (0.578 + 1.00i)4-s + (6.77 + 11.7i)5-s + (−5.29 − 12.5i)6-s + (3.5 − 6.06i)7-s − 23.9·8-s + (−7.38 − 25.9i)9-s − 35.4·10-s + (−12.0 + 20.8i)11-s + (−5.97 − 0.740i)12-s + (−11.9 − 20.7i)13-s + (9.15 + 15.8i)14-s + (−69.8 − 8.65i)15-s + (26.6 − 46.2i)16-s + 79.6·17-s + ⋯
 L(s)  = 1 + (−0.462 + 0.800i)2-s + (−0.602 + 0.797i)3-s + (0.0723 + 0.125i)4-s + (0.605 + 1.04i)5-s + (−0.360 − 0.851i)6-s + (0.188 − 0.327i)7-s − 1.05·8-s + (−0.273 − 0.961i)9-s − 1.12·10-s + (−0.329 + 0.571i)11-s + (−0.143 − 0.0178i)12-s + (−0.255 − 0.443i)13-s + (0.174 + 0.302i)14-s + (−1.20 − 0.149i)15-s + (0.417 − 0.722i)16-s + 1.13·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.994 + 0.102i$ Motivic weight: $$3$$ Character: $\chi_{63} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ -0.994 + 0.102i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.0459966 - 0.897587i$$ $$L(\frac12)$$ $$\approx$$ $$0.0459966 - 0.897587i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (3.13 - 4.14i)T$$
7 $$1 + (-3.5 + 6.06i)T$$
good2 $$1 + (1.30 - 2.26i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (-6.77 - 11.7i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (12.0 - 20.8i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (11.9 + 20.7i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 - 79.6T + 4.91e3T^{2}$$
19 $$1 + 50.0T + 6.85e3T^{2}$$
23 $$1 + (-75.8 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (128. - 221. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-1.36 - 2.35i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 319.T + 5.06e4T^{2}$$
41 $$1 + (82.3 + 142. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (211. - 366. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (50.0 - 86.6i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 194.T + 1.48e5T^{2}$$
59 $$1 + (288. + 499. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-21.5 + 37.2i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-508. - 880. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 509.T + 3.57e5T^{2}$$
73 $$1 - 1.03e3T + 3.89e5T^{2}$$
79 $$1 + (-447. + 774. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (7.23 - 12.5i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 1.53e3T + 7.04e5T^{2}$$
97 $$1 + (-887. + 1.53e3i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$