Properties

Degree 2
Conductor 3
Sign $0.735 - 0.677i$
Motivic weight 38
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.26e5i·2-s + (−8.54e8 + 7.87e8i)3-s + 2.23e11·4-s + 6.89e12i·5-s + (−1.78e14 − 1.93e14i)6-s + 1.19e16·7-s + 1.12e17i·8-s + (1.09e17 − 1.34e18i)9-s − 1.55e18·10-s − 1.21e20i·11-s + (−1.91e20 + 1.76e20i)12-s + 1.57e21·13-s + 2.70e21i·14-s + (−5.42e21 − 5.88e21i)15-s + 3.60e22·16-s − 2.37e23i·17-s + ⋯
L(s)  = 1  + 0.431i·2-s + (−0.735 + 0.677i)3-s + 0.814·4-s + 0.361i·5-s + (−0.292 − 0.317i)6-s + 1.05·7-s + 0.782i·8-s + (0.0813 − 0.996i)9-s − 0.155·10-s − 1.98i·11-s + (−0.598 + 0.551i)12-s + 1.07·13-s + 0.453i·14-s + (−0.244 − 0.265i)15-s + 0.476·16-s − 0.993i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $0.735 - 0.677i$
motivic weight  =  \(38\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :19),\ 0.735 - 0.677i)\)
\(L(\frac{39}{2})\)  \(\approx\)  \(2.306801110\)
\(L(\frac12)\)  \(\approx\)  \(2.306801110\)
\(L(20)\)   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 \( 1 + (8.54e8 - 7.87e8i)T \)
good2 \( 1 - 2.26e5iT - 2.74e11T^{2} \)
5 \( 1 - 6.89e12iT - 3.63e26T^{2} \)
7 \( 1 - 1.19e16T + 1.29e32T^{2} \)
11 \( 1 + 1.21e20iT - 3.74e39T^{2} \)
13 \( 1 - 1.57e21T + 2.13e42T^{2} \)
17 \( 1 + 2.37e23iT - 5.71e46T^{2} \)
19 \( 1 + 3.40e23T + 3.91e48T^{2} \)
23 \( 1 - 1.60e25iT - 5.56e51T^{2} \)
29 \( 1 - 5.04e27iT - 3.72e55T^{2} \)
31 \( 1 - 1.48e28T + 4.69e56T^{2} \)
37 \( 1 - 4.44e29T + 3.90e59T^{2} \)
41 \( 1 + 2.53e30iT - 1.93e61T^{2} \)
43 \( 1 + 1.36e31T + 1.17e62T^{2} \)
47 \( 1 - 1.20e31iT - 3.46e63T^{2} \)
53 \( 1 + 1.42e32iT - 3.33e65T^{2} \)
59 \( 1 + 5.18e33iT - 1.96e67T^{2} \)
61 \( 1 + 1.25e34T + 6.95e67T^{2} \)
67 \( 1 - 7.45e34T + 2.45e69T^{2} \)
71 \( 1 - 2.70e35iT - 2.22e70T^{2} \)
73 \( 1 - 1.76e35T + 6.40e70T^{2} \)
79 \( 1 - 4.16e34T + 1.28e72T^{2} \)
83 \( 1 - 4.66e35iT - 8.41e72T^{2} \)
89 \( 1 - 1.44e37iT - 1.19e74T^{2} \)
97 \( 1 - 4.27e37T + 3.14e75T^{2} \)
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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

−16.73585599166079981893635829105, −15.74472562804191976410538802401, −14.23289677370193367801389485054, −11.41451616805831421631664883538, −10.92082739950430188207466530437, −8.430665765968341175966966996219, −6.48616495129974343430905921859, −5.28953919309570227897402305380, −3.22418553489561592771511727198, −1.00036783132926991728984195178, 1.23373382131977621061706890140, 2.01777581639893428985006130400, 4.60077624233836130690208887898, 6.42544803705609877483655362668, 7.86822300057887383423741248151, 10.47897941182940439250343351277, 11.74653574032614817029694523472, 12.85050338515043644192853441239, 15.17276045181292562320720653788, 16.96781366026417444569755219139

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