Properties

Degree 2
Conductor 3
Sign $0.555 - 0.831i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22.4i·2-s + (45 − 67.3i)3-s − 248·4-s − 224. i·5-s + (1.51e3 + 1.01e3i)6-s − 1.75e3·7-s + 179. i·8-s + (−2.51e3 − 6.06e3i)9-s + 5.04e3·10-s + 6.95e3i·11-s + (−1.11e4 + 1.67e4i)12-s + 2.57e4·13-s − 3.92e4i·14-s + (−1.51e4 − 1.01e4i)15-s − 6.75e4·16-s + 7.48e4i·17-s + ⋯
L(s)  = 1  + 1.40i·2-s + (0.555 − 0.831i)3-s − 0.968·4-s − 0.359i·5-s + (1.16 + 0.779i)6-s − 0.728·7-s + 0.0438i·8-s + (−0.382 − 0.923i)9-s + 0.504·10-s + 0.475i·11-s + (−0.538 + 0.805i)12-s + 0.900·13-s − 1.02i·14-s + (−0.298 − 0.199i)15-s − 1.03·16-s + 0.896i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $0.555 - 0.831i$
motivic weight  =  \(8\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :4),\ 0.555 - 0.831i)$
$L(\frac{9}{2})$  $\approx$  $1.03711 + 0.554359i$
$L(\frac12)$  $\approx$  $1.03711 + 0.554359i$
$L(5)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (-45 + 67.3i)T \)
good2 \( 1 - 22.4iT - 256T^{2} \)
5 \( 1 + 224. iT - 3.90e5T^{2} \)
7 \( 1 + 1.75e3T + 5.76e6T^{2} \)
11 \( 1 - 6.95e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.57e4T + 8.15e8T^{2} \)
17 \( 1 - 7.48e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.89e4T + 1.69e10T^{2} \)
23 \( 1 + 4.70e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.60e5iT - 5.00e11T^{2} \)
31 \( 1 + 3.51e5T + 8.52e11T^{2} \)
37 \( 1 - 1.33e6T + 3.51e12T^{2} \)
41 \( 1 - 1.87e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.52e6T + 1.16e13T^{2} \)
47 \( 1 + 4.08e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.60e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.37e7iT - 1.46e14T^{2} \)
61 \( 1 - 7.53e5T + 1.91e14T^{2} \)
67 \( 1 - 2.26e6T + 4.06e14T^{2} \)
71 \( 1 - 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.76e7T + 8.06e14T^{2} \)
79 \( 1 + 2.29e7T + 1.51e15T^{2} \)
83 \( 1 + 4.63e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.26e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.47e8T + 7.83e15T^{2} \)
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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

−25.43404712791207914444593543883, −24.24988930999681135601755894035, −23.03125818036933381721165597225, −20.23187960048157825678945100376, −18.29684445410031745836940228720, −16.52349202435110678096070736482, −14.76513444245845090232475090568, −12.97742864252240100208929908279, −8.537722471016272724437647432597, −6.56108305736119635078666434601, 3.29129434150346804494000997655, 9.509896719961398489491287140507, 11.13538667100853805576196445480, 13.54027667979888221166920257283, 15.90659526368572752200092385994, 18.82264965358014825023913936402, 20.12216169702305363155548438682, 21.40363457458476111200117821823, 22.61633771030336282354266566175, 25.48457455056030862586971118301

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