Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.666 - 0.745i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.53 + 1.28i)2-s + (−2.46 + 4.57i)3-s + (4.69 + 3.93i)4-s + (1.06 − 6.01i)5-s + (−14.5 + 12.9i)6-s + (13.2 − 11.0i)7-s + (−3.52 − 6.11i)8-s + (−14.7 − 22.5i)9-s + (11.4 − 19.8i)10-s + (6.89 + 39.1i)11-s + (−29.5 + 11.7i)12-s + (−25.7 + 9.35i)13-s + (60.9 − 22.1i)14-s + (24.8 + 19.6i)15-s + (−13.1 − 74.3i)16-s + (−54.8 + 95.0i)17-s + ⋯
L(s)  = 1  + (1.24 + 0.454i)2-s + (−0.475 + 0.879i)3-s + (0.586 + 0.491i)4-s + (0.0948 − 0.537i)5-s + (−0.993 + 0.882i)6-s + (0.713 − 0.598i)7-s + (−0.155 − 0.270i)8-s + (−0.548 − 0.836i)9-s + (0.362 − 0.628i)10-s + (0.189 + 1.07i)11-s + (−0.711 + 0.281i)12-s + (−0.548 + 0.199i)13-s + (1.16 − 0.423i)14-s + (0.428 + 0.339i)15-s + (−0.204 − 1.16i)16-s + (−0.782 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.666 - 0.745i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.666 - 0.745i)$
$L(2)$  $\approx$  $1.63733 + 0.732199i$
$L(\frac12)$  $\approx$  $1.63733 + 0.732199i$
$L(\frac{5}{2})$   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 \( 1 + (2.46 - 4.57i)T \)
good2 \( 1 + (-3.53 - 1.28i)T + (6.12 + 5.14i)T^{2} \)
5 \( 1 + (-1.06 + 6.01i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (-13.2 + 11.0i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (-6.89 - 39.1i)T + (-1.25e3 + 455. i)T^{2} \)
13 \( 1 + (25.7 - 9.35i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (54.8 - 95.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (61.0 + 105. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-96.2 - 80.7i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (104. + 37.9i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (-148. - 124. i)T + (5.17e3 + 2.93e4i)T^{2} \)
37 \( 1 + (-46.0 + 79.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (103. - 37.5i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (74.3 + 421. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (77.3 - 64.9i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 - 338.T + 1.48e5T^{2} \)
59 \( 1 + (54.4 - 309. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (363. - 304. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (-299. + 108. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-385. + 668. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (274. + 475. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-972. - 354. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-195. - 71.2i)T + (4.38e5 + 3.67e5i)T^{2} \)
89 \( 1 + (312. + 540. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-69.8 - 396. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−16.91402950414158629946468695447, −15.29932397590622468743996638664, −14.89244539092692497174910189879, −13.37622475714253896923672889933, −12.21275284740608602901160656383, −10.74748276992125709032849744975, −9.150386082435377059729470487040, −6.82065775857669751174734112222, −5.04106116616245036173610261640, −4.25996973471083041514066543281, 2.59276614497512186505977617562, 5.06458106877346768904953092051, 6.41703346438527415172558337267, 8.360756290991229942867177722041, 11.00636933420986866389778052841, 11.75555839988264702070612059847, 12.90568993801523602492638655269, 14.02646603161040059170224120270, 14.86399624106468834961190208955, 16.76350495991083504132332108563

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