Properties

Degree 2
Conductor 3
Sign $-0.672 - 0.739i$
Motivic weight 38
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.17e5i·2-s + (7.82e8 + 8.59e8i)3-s + 2.27e11·4-s − 1.16e13i·5-s + (−1.86e14 + 1.69e14i)6-s − 8.59e15·7-s + 1.09e17i·8-s + (−1.27e17 + 1.34e18i)9-s + 2.52e18·10-s + 3.26e19i·11-s + (1.78e20 + 1.95e20i)12-s − 6.39e20·13-s − 1.86e21i·14-s + (9.98e21 − 9.08e21i)15-s + 3.89e22·16-s + 3.38e23i·17-s + ⋯
L(s)  = 1  + 0.414i·2-s + (0.672 + 0.739i)3-s + 0.828·4-s − 0.608i·5-s + (−0.306 + 0.278i)6-s − 0.754·7-s + 0.757i·8-s + (−0.0942 + 0.995i)9-s + 0.252·10-s + 0.533i·11-s + (0.557 + 0.612i)12-s − 0.437·13-s − 0.312i·14-s + (0.450 − 0.409i)15-s + 0.515·16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-0.672 - 0.739i$
motivic weight  =  \(38\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :19),\ -0.672 - 0.739i)\)
\(L(\frac{39}{2})\)  \(\approx\)  \(2.448444305\)
\(L(\frac12)\)  \(\approx\)  \(2.448444305\)
\(L(20)\)   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 \( 1 + (-7.82e8 - 8.59e8i)T \)
good2 \( 1 - 2.17e5iT - 2.74e11T^{2} \)
5 \( 1 + 1.16e13iT - 3.63e26T^{2} \)
7 \( 1 + 8.59e15T + 1.29e32T^{2} \)
11 \( 1 - 3.26e19iT - 3.74e39T^{2} \)
13 \( 1 + 6.39e20T + 2.13e42T^{2} \)
17 \( 1 - 3.38e23iT - 5.71e46T^{2} \)
19 \( 1 - 9.26e23T + 3.91e48T^{2} \)
23 \( 1 - 5.58e25iT - 5.56e51T^{2} \)
29 \( 1 - 8.27e27iT - 3.72e55T^{2} \)
31 \( 1 + 3.97e28T + 4.69e56T^{2} \)
37 \( 1 - 2.19e28T + 3.90e59T^{2} \)
41 \( 1 + 8.17e30iT - 1.93e61T^{2} \)
43 \( 1 - 1.17e31T + 1.17e62T^{2} \)
47 \( 1 + 8.05e31iT - 3.46e63T^{2} \)
53 \( 1 - 7.18e32iT - 3.33e65T^{2} \)
59 \( 1 + 4.73e32iT - 1.96e67T^{2} \)
61 \( 1 - 8.08e33T + 6.95e67T^{2} \)
67 \( 1 - 1.06e34T + 2.45e69T^{2} \)
71 \( 1 + 1.81e34iT - 2.22e70T^{2} \)
73 \( 1 + 1.52e35T + 6.40e70T^{2} \)
79 \( 1 - 1.40e36T + 1.28e72T^{2} \)
83 \( 1 + 2.75e36iT - 8.41e72T^{2} \)
89 \( 1 - 1.01e37iT - 1.19e74T^{2} \)
97 \( 1 - 8.94e37T + 3.14e75T^{2} \)
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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

−16.90183544187103795249953031142, −15.85295505786638307726516143085, −14.65684635077864202098718929542, −12.64101751760180327077053271707, −10.55094518596279611282901119293, −8.976426893281571568677219716718, −7.32778407755043976703071045690, −5.39926354486135136557540238146, −3.52676726717133813511835211618, −1.91127159309978392584790721650, 0.67961730440859792950145733451, 2.43063665718798610214224573223, 3.24471855439660385653570281682, 6.37989444611203857802791601417, 7.46391611601850010566088902556, 9.603838084563848302835922119083, 11.41727979842045022962530374951, 12.88045493133187469748060821721, 14.51615718531865624495231801434, 16.16196844892552496788827826379

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