# Properties

 Label 2-363-33.17-c1-0-10 Degree $2$ Conductor $363$ Sign $0.958 - 0.285i$ Analytic cond. $2.89856$ Root an. cond. $1.70251$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.37 + 1.04i)3-s + (1.61 − 1.17i)4-s + (3.15 + 1.02i)5-s + (0.804 − 2.89i)9-s + (−1.00 + 3.31i)12-s + (−5.42 + 1.89i)15-s + (1.23 − 3.80i)16-s + (6.30 − 2.04i)20-s + 3.31i·23-s + (4.85 + 3.52i)25-s + (1.91 + 4.82i)27-s + (1.54 + 4.75i)31-s + (−2.09 − 5.62i)36-s + (5.66 − 4.11i)37-s + (5.5 − 8.29i)45-s + ⋯
 L(s)  = 1 + (−0.796 + 0.604i)3-s + (0.809 − 0.587i)4-s + (1.41 + 0.458i)5-s + (0.268 − 0.963i)9-s + (−0.288 + 0.957i)12-s + (−1.40 + 0.488i)15-s + (0.309 − 0.951i)16-s + (1.41 − 0.458i)20-s + 0.691i·23-s + (0.970 + 0.705i)25-s + (0.369 + 0.929i)27-s + (0.277 + 0.854i)31-s + (−0.349 − 0.937i)36-s + (0.931 − 0.676i)37-s + (0.819 − 1.23i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$363$$    =    $$3 \cdot 11^{2}$$ Sign: $0.958 - 0.285i$ Analytic conductor: $$2.89856$$ Root analytic conductor: $$1.70251$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{363} (215, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 363,\ (\ :1/2),\ 0.958 - 0.285i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.54794 + 0.225814i$$ $$L(\frac12)$$ $$\approx$$ $$1.54794 + 0.225814i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.37 - 1.04i)T$$
11 $$1$$
good2 $$1 + (-1.61 + 1.17i)T^{2}$$
5 $$1 + (-3.15 - 1.02i)T + (4.04 + 2.93i)T^{2}$$
7 $$1 + (-2.16 + 6.65i)T^{2}$$
13 $$1 + (10.5 - 7.64i)T^{2}$$
17 $$1 + (-13.7 - 9.99i)T^{2}$$
19 $$1 + (-5.87 - 18.0i)T^{2}$$
23 $$1 - 3.31iT - 23T^{2}$$
29 $$1 + (8.96 - 27.5i)T^{2}$$
31 $$1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2}$$
37 $$1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2}$$
41 $$1 + (12.6 + 38.9i)T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + (3.89 - 5.36i)T + (-14.5 - 44.6i)T^{2}$$
53 $$1 + (12.6 - 4.09i)T + (42.8 - 31.1i)T^{2}$$
59 $$1 + (1.94 + 2.68i)T + (-18.2 + 56.1i)T^{2}$$
61 $$1 + (49.3 + 35.8i)T^{2}$$
67 $$1 + 13T + 67T^{2}$$
71 $$1 + (15.7 + 5.12i)T + (57.4 + 41.7i)T^{2}$$
73 $$1 + (-22.5 + 69.4i)T^{2}$$
79 $$1 + (63.9 - 46.4i)T^{2}$$
83 $$1 + (-67.1 - 48.7i)T^{2}$$
89 $$1 + 16.5iT - 89T^{2}$$
97 $$1 + (-5.25 - 16.1i)T + (-78.4 + 57.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$