Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.710 + 0.703i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.404 − 2.29i)2-s + (4.52 + 2.54i)3-s + (2.41 − 0.878i)4-s + (−4.78 − 4.01i)5-s + (4.01 − 11.4i)6-s + (2.53 + 0.924i)7-s + (−12.3 − 21.3i)8-s + (14.0 + 23.0i)9-s + (−7.27 + 12.5i)10-s + (−30.2 + 25.4i)11-s + (13.1 + 2.16i)12-s + (−12.8 + 73.0i)13-s + (1.09 − 6.20i)14-s + (−11.4 − 30.3i)15-s + (−28.2 + 23.6i)16-s + (29.1 − 50.4i)17-s + ⋯
L(s)  = 1  + (−0.143 − 0.811i)2-s + (0.871 + 0.490i)3-s + (0.301 − 0.109i)4-s + (−0.427 − 0.358i)5-s + (0.273 − 0.777i)6-s + (0.137 + 0.0499i)7-s + (−0.544 − 0.942i)8-s + (0.519 + 0.854i)9-s + (−0.230 + 0.398i)10-s + (−0.830 + 0.696i)11-s + (0.316 + 0.0521i)12-s + (−0.274 + 1.55i)13-s + (0.0208 − 0.118i)14-s + (−0.196 − 0.522i)15-s + (−0.441 + 0.370i)16-s + (0.415 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.710 + 0.703i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.710 + 0.703i)$
$L(2)$  $\approx$  $1.29197 - 0.531602i$
$L(\frac12)$  $\approx$  $1.29197 - 0.531602i$
$L(\frac{5}{2})$   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 \( 1 + (-4.52 - 2.54i)T \)
good2 \( 1 + (0.404 + 2.29i)T + (-7.51 + 2.73i)T^{2} \)
5 \( 1 + (4.78 + 4.01i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-2.53 - 0.924i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (30.2 - 25.4i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (12.8 - 73.0i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-29.1 + 50.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (41.2 + 71.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (25.8 - 9.40i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-30.1 - 170. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-149. + 54.3i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-220. + 382. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-14.7 + 83.6i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (150. - 126. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-402. - 146. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 - 448.T + 1.48e5T^{2} \)
59 \( 1 + (269. + 225. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (346. + 126. i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (104. - 594. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (423. - 733. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-21.1 - 36.7i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-63.6 - 361. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (10.0 + 57.0i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (713. + 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-875. + 735. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−16.34458057818370827967222014259, −15.54413281341343332188954517563, −14.28166825296667958008548125367, −12.73904848025487104542606885938, −11.48204488871716694912480487389, −10.12985707350246756818272109201, −9.009842261561972835557430707373, −7.27721143077731023880240813839, −4.42828702090385129633570218169, −2.39867116015528794661633291394, 3.01147425514817567900991837094, 6.05470829923183223373488274354, 7.76832832566978922344516004799, 8.200698706852458310524827326397, 10.41772185693323383850905500246, 12.11667530248520477050476078919, 13.49229317567728086699932102075, 14.99096038111506856952477459243, 15.39536513168631465414180335028, 16.92750191511050241737774508716

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