Properties

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

Origins

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Normalization:  

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

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