# Properties

 Label 2-588-21.17-c3-0-27 Degree $2$ Conductor $588$ Sign $0.997 + 0.0633i$ Analytic cond. $34.6931$ Root an. cond. $5.89008$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.5 + 2.59i)3-s + (13.5 + 23.3i)9-s − 91.7i·13-s + (109.5 − 63.2i)19-s + (62.5 − 108. i)25-s + 140. i·27-s + (163.5 + 94.3i)31-s + (161.5 + 279. i)37-s + (238.5 − 413. i)39-s − 71·43-s + 657·57-s + (810 − 467. i)61-s + (63.5 − 109. i)67-s + (−1.05e3 − 608. i)73-s + (562.5 − 324. i)75-s + ⋯
 L(s)  = 1 + (0.866 + 0.499i)3-s + (0.5 + 0.866i)9-s − 1.95i·13-s + (1.32 − 0.763i)19-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (0.947 + 0.546i)31-s + (0.717 + 1.24i)37-s + (0.979 − 1.69i)39-s − 0.251·43-s + 1.52·57-s + (1.70 − 0.981i)61-s + (0.115 − 0.200i)67-s + (−1.69 − 0.976i)73-s + (0.866 − 0.499i)75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$588$$    =    $$2^{2} \cdot 3 \cdot 7^{2}$$ Sign: $0.997 + 0.0633i$ Analytic conductor: $$34.6931$$ Root analytic conductor: $$5.89008$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{588} (521, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 588,\ (\ :3/2),\ 0.997 + 0.0633i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.881265942$$ $$L(\frac12)$$ $$\approx$$ $$2.881265942$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-4.5 - 2.59i)T$$
7 $$1$$
good5 $$1 + (-62.5 + 108. i)T^{2}$$
11 $$1 + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 91.7iT - 2.19e3T^{2}$$
17 $$1 + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-109.5 + 63.2i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + (-163.5 - 94.3i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-161.5 - 279. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 + 71T + 7.95e4T^{2}$$
47 $$1 + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-810 + 467. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-63.5 + 109. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 3.57e5T^{2}$$
73 $$1 + (1.05e3 + 608. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-693.5 - 1.20e3i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 1.37e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$