# Properties

 Degree 2 Conductor 3 Sign $0.381 - 0.924i$ Motivic weight 38 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + 9.77e5i·2-s + (−4.43e8 + 1.07e9i)3-s − 6.80e11·4-s − 1.94e13i·5-s + (−1.05e15 − 4.33e14i)6-s − 1.85e15·7-s − 3.96e17i·8-s + (−9.57e17 − 9.52e17i)9-s + 1.89e19·10-s + 1.00e20i·11-s + (3.01e20 − 7.31e20i)12-s + 5.34e20·13-s − 1.81e21i·14-s + (2.08e22 + 8.60e21i)15-s + 2.00e23·16-s − 2.93e23i·17-s + ⋯
 L(s)  = 1 + 1.86i·2-s + (−0.381 + 0.924i)3-s − 2.47·4-s − 1.01i·5-s + (−1.72 − 0.711i)6-s − 0.162·7-s − 2.75i·8-s + (−0.708 − 0.705i)9-s + 1.89·10-s + 1.64i·11-s + (0.944 − 2.28i)12-s + 0.365·13-s − 0.303i·14-s + (0.940 + 0.388i)15-s + 2.65·16-s − 1.22i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(39-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $0.381 - 0.924i$ motivic weight = $$38$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3,\ (\ :19),\ 0.381 - 0.924i)$$ $$L(\frac{39}{2})$$ $$\approx$$ $$0.7796303641$$ $$L(\frac12)$$ $$\approx$$ $$0.7796303641$$ $$L(20)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (4.43e8 - 1.07e9i)T$$
good2 $$1 - 9.77e5iT - 2.74e11T^{2}$$
5 $$1 + 1.94e13iT - 3.63e26T^{2}$$
7 $$1 + 1.85e15T + 1.29e32T^{2}$$
11 $$1 - 1.00e20iT - 3.74e39T^{2}$$
13 $$1 - 5.34e20T + 2.13e42T^{2}$$
17 $$1 + 2.93e23iT - 5.71e46T^{2}$$
19 $$1 + 2.13e24T + 3.91e48T^{2}$$
23 $$1 + 6.81e25iT - 5.56e51T^{2}$$
29 $$1 - 4.43e27iT - 3.72e55T^{2}$$
31 $$1 - 9.20e27T + 4.69e56T^{2}$$
37 $$1 + 1.71e29T + 3.90e59T^{2}$$
41 $$1 - 3.25e30iT - 1.93e61T^{2}$$
43 $$1 - 9.05e30T + 1.17e62T^{2}$$
47 $$1 + 4.90e31iT - 3.46e63T^{2}$$
53 $$1 + 3.19e32iT - 3.33e65T^{2}$$
59 $$1 + 2.55e33iT - 1.96e67T^{2}$$
61 $$1 - 1.24e34T + 6.95e67T^{2}$$
67 $$1 - 4.69e34T + 2.45e69T^{2}$$
71 $$1 + 1.03e35iT - 2.22e70T^{2}$$
73 $$1 + 1.12e35T + 6.40e70T^{2}$$
79 $$1 - 5.18e34T + 1.28e72T^{2}$$
83 $$1 + 1.87e35iT - 8.41e72T^{2}$$
89 $$1 + 4.77e36iT - 1.19e74T^{2}$$
97 $$1 + 7.67e37T + 3.14e75T^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

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

−16.74963244862995665474904043345, −15.82197135861849333499468936572, −14.62542641510444856928757754251, −12.69396066589494154531868025355, −9.715147753258901654004919081275, −8.586743278459338278942291007680, −6.72517648669712767291706718509, −5.09368767650786243154082479072, −4.38906143409704775227665208427, −0.34560847179440494012829232001, 0.990355989494126016923096730746, 2.38944076023095451840585459871, 3.58642711543248984438273543662, 6.02639871413492288459843980354, 8.442103075816099858598083690510, 10.62205826429000351356473636992, 11.36789365005760385118219804586, 12.92274480096132654651161289204, 14.05028027162195665902483114213, 17.46926796807307555229277098780