Properties

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

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line