# Properties

 Degree $2$ Conductor $3$ Sign $0.851 - 0.524i$ Motivic weight $38$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 6.48e5i·2-s + (−9.89e8 + 6.09e8i)3-s − 1.45e11·4-s − 1.81e13i·5-s + (3.95e14 + 6.41e14i)6-s − 1.77e16·7-s − 8.40e16i·8-s + (6.07e17 − 1.20e18i)9-s − 1.17e19·10-s − 1.89e19i·11-s + (1.43e20 − 8.84e19i)12-s − 1.72e21·13-s + 1.15e22i·14-s + (1.10e22 + 1.79e22i)15-s − 9.43e22·16-s + 2.70e23i·17-s + ⋯
 L(s)  = 1 − 1.23i·2-s + (−0.851 + 0.524i)3-s − 0.528·4-s − 0.952i·5-s + (0.648 + 1.05i)6-s − 1.55·7-s − 0.583i·8-s + (0.449 − 0.893i)9-s − 1.17·10-s − 0.309i·11-s + (0.449 − 0.277i)12-s − 1.18·13-s + 1.92i·14-s + (0.499 + 0.810i)15-s − 1.24·16-s + 1.13i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(39-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3$$ Sign: $0.851 - 0.524i$ Motivic weight: $$38$$ Character: $\chi_{3} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3,\ (\ :19),\ 0.851 - 0.524i)$$

## Particular Values

 $$L(\frac{39}{2})$$ $$\approx$$ $$0.1896044109$$ $$L(\frac12)$$ $$\approx$$ $$0.1896044109$$ $$L(20)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (9.89e8 - 6.09e8i)T$$
good2 $$1 + 6.48e5iT - 2.74e11T^{2}$$
5 $$1 + 1.81e13iT - 3.63e26T^{2}$$
7 $$1 + 1.77e16T + 1.29e32T^{2}$$
11 $$1 + 1.89e19iT - 3.74e39T^{2}$$
13 $$1 + 1.72e21T + 2.13e42T^{2}$$
17 $$1 - 2.70e23iT - 5.71e46T^{2}$$
19 $$1 - 1.47e24T + 3.91e48T^{2}$$
23 $$1 + 7.60e25iT - 5.56e51T^{2}$$
29 $$1 + 5.84e26iT - 3.72e55T^{2}$$
31 $$1 - 3.31e28T + 4.69e56T^{2}$$
37 $$1 + 2.91e29T + 3.90e59T^{2}$$
41 $$1 - 3.49e30iT - 1.93e61T^{2}$$
43 $$1 + 3.83e30T + 1.17e62T^{2}$$
47 $$1 - 8.27e31iT - 3.46e63T^{2}$$
53 $$1 - 1.02e33iT - 3.33e65T^{2}$$
59 $$1 + 7.44e33iT - 1.96e67T^{2}$$
61 $$1 + 1.73e33T + 6.95e67T^{2}$$
67 $$1 + 4.97e34T + 2.45e69T^{2}$$
71 $$1 + 2.73e34iT - 2.22e70T^{2}$$
73 $$1 - 2.76e35T + 6.40e70T^{2}$$
79 $$1 + 1.24e36T + 1.28e72T^{2}$$
83 $$1 + 4.54e36iT - 8.41e72T^{2}$$
89 $$1 + 1.42e36iT - 1.19e74T^{2}$$
97 $$1 + 3.92e37T + 3.14e75T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−16.93610713106367878908739905728, −15.84267564248942384197086112567, −12.83020310676957650431627557835, −12.12726272589125032542873551732, −10.35735574046018170185370271560, −9.396790988772986170528913578110, −6.36381320836944933483108150784, −4.51129165117027970763172250465, −3.04357356215223300359996580863, −0.972892460934494412561330724251, 0.083869322673026564294775701764, 2.69440408607607329674589574666, 5.32918657279288121467588468372, 6.73763582982972789692531549899, 7.24618448281613907258187732011, 9.933415673092850691673745189179, 11.86710591706813783300926981046, 13.71113985333961489712984557002, 15.48228136003490638746281602060, 16.62943324420727998805548903507