Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.789 - 0.613i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.644 − 0.540i)2-s + (0.739 + 5.14i)3-s + (−1.26 + 7.18i)4-s + (10.1 − 3.69i)5-s + (3.25 + 2.91i)6-s + (−4.63 − 26.2i)7-s + (6.43 + 11.1i)8-s + (−25.9 + 7.60i)9-s + (4.54 − 7.87i)10-s + (17.1 + 6.22i)11-s + (−37.8 − 1.20i)12-s + (−27.1 − 22.7i)13-s + (−17.1 − 14.4i)14-s + (26.5 + 49.5i)15-s + (−44.6 − 16.2i)16-s + (56.4 − 97.8i)17-s + ⋯
L(s)  = 1  + (0.227 − 0.191i)2-s + (0.142 + 0.989i)3-s + (−0.158 + 0.897i)4-s + (0.909 − 0.330i)5-s + (0.221 + 0.198i)6-s + (−0.250 − 1.41i)7-s + (0.284 + 0.492i)8-s + (−0.959 + 0.281i)9-s + (0.143 − 0.249i)10-s + (0.469 + 0.170i)11-s + (−0.911 − 0.0289i)12-s + (−0.578 − 0.485i)13-s + (−0.328 − 0.275i)14-s + (0.456 + 0.852i)15-s + (−0.697 − 0.253i)16-s + (0.805 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.789 - 0.613i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.789 - 0.613i)$
$L(2)$  $\approx$  $1.32056 + 0.453148i$
$L(\frac12)$  $\approx$  $1.32056 + 0.453148i$
$L(\frac{5}{2})$   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 \( 1 + (-0.739 - 5.14i)T \)
good2 \( 1 + (-0.644 + 0.540i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (-10.1 + 3.69i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (4.63 + 26.2i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (-17.1 - 6.22i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (27.1 + 22.7i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-56.4 + 97.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.33 - 16.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (24.1 - 137. i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-61.6 + 51.7i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (42.0 - 238. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (16.7 - 29.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (297. + 249. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (79.2 + 28.8i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-80.1 - 454. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 - 351.T + 1.48e5T^{2} \)
59 \( 1 + (47.3 - 17.2i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (24.6 + 139. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (79.8 + 67.0i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-161. + 280. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (292. + 506. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (486. - 408. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-740. + 621. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-233. - 403. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.56e3 - 571. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−17.01790441042041029220654668973, −16.05874410039137570704918106768, −14.13852835921140285163427033145, −13.52974881548668990364157101514, −11.91124368758628129859292720846, −10.29385727416662643969242323626, −9.305275267733759671334075690395, −7.49396701856777052585476221376, −5.02499560056153580832387316024, −3.44469370570110204412695572750, 2.08666146997398087738577047962, 5.71386263896642540627790974466, 6.49013171186753472735014375868, 8.726053700375607856107708513663, 10.02461366457003226825794705308, 11.89850315260074013003218359751, 13.11453715951536114781206193746, 14.35824754064933989509947668013, 14.99152616658038698811336340778, 16.88554678200133468740048887138

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