# Properties

 Label 2-63-7.2-c3-0-5 Degree $2$ Conductor $63$ Sign $0.998 + 0.0589i$ Analytic cond. $3.71712$ Root an. cond. $1.92798$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.27 + 3.94i)2-s + (−6.38 − 11.0i)4-s + (8.93 − 15.4i)5-s + (2.26 − 18.3i)7-s + 21.6·8-s + (40.7 + 70.5i)10-s + (−5.69 − 9.86i)11-s − 13.0·13-s + (67.3 + 50.7i)14-s + (1.62 − 2.81i)16-s + (26.6 + 46.1i)17-s + (21.2 − 36.7i)19-s − 228.·20-s + 51.9·22-s + (76.0 − 131. i)23-s + ⋯
 L(s)  = 1 + (−0.805 + 1.39i)2-s + (−0.797 − 1.38i)4-s + (0.799 − 1.38i)5-s + (0.122 − 0.992i)7-s + 0.958·8-s + (1.28 + 2.23i)10-s + (−0.156 − 0.270i)11-s − 0.279·13-s + (1.28 + 0.969i)14-s + (0.0254 − 0.0440i)16-s + (0.379 + 0.658i)17-s + (0.256 − 0.443i)19-s − 2.54·20-s + 0.503·22-s + (0.689 − 1.19i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.998 + 0.0589i$ Analytic conductor: $$3.71712$$ Root analytic conductor: $$1.92798$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{63} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.998 + 0.0589i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.940238 - 0.0277375i$$ $$L(\frac12)$$ $$\approx$$ $$0.940238 - 0.0277375i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (-2.26 + 18.3i)T$$
good2 $$1 + (2.27 - 3.94i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (-8.93 + 15.4i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (5.69 + 9.86i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 13.0T + 2.19e3T^{2}$$
17 $$1 + (-26.6 - 46.1i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-21.2 + 36.7i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-76.0 + 131. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 186.T + 2.43e4T^{2}$$
31 $$1 + (-78.9 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (1.87 - 3.24i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 39.3T + 6.89e4T^{2}$$
43 $$1 - 429.T + 7.95e4T^{2}$$
47 $$1 + (-10.5 + 18.3i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-182. - 316. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (325. - 564. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (72.7 + 125. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 368.T + 3.57e5T^{2}$$
73 $$1 + (304. + 527. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (455. - 788. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 327.T + 5.71e5T^{2}$$
89 $$1 + (18.8 - 32.5i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 722.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$