Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.781 + 0.624i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.30 + 2.65i)2-s + (−0.728 + 15.5i)3-s + (21.7 − 18.2i)4-s + (15.2 + 86.2i)5-s + (−36.0 − 115. i)6-s + (−90.4 − 75.9i)7-s + (14.1 − 24.4i)8-s + (−241. − 22.7i)9-s + (−340. − 589. i)10-s + (19.6 − 111. i)11-s + (268. + 351. i)12-s + (−178. − 64.9i)13-s + (862. + 313. i)14-s + (−1.35e3 + 173. i)15-s + (−195. + 1.10e3i)16-s + (522. + 905. i)17-s + ⋯
L(s)  = 1  + (−1.29 + 0.469i)2-s + (−0.0467 + 0.998i)3-s + (0.679 − 0.569i)4-s + (0.272 + 1.54i)5-s + (−0.408 − 1.31i)6-s + (−0.698 − 0.585i)7-s + (0.0779 − 0.134i)8-s + (−0.995 − 0.0934i)9-s + (−1.07 − 1.86i)10-s + (0.0490 − 0.278i)11-s + (0.537 + 0.705i)12-s + (−0.292 − 0.106i)13-s + (1.17 + 0.428i)14-s + (−1.55 + 0.199i)15-s + (−0.191 + 1.08i)16-s + (0.438 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.781 + 0.624i$
motivic weight  =  \(5\)
character  :  $\chi_{27} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :5/2),\ -0.781 + 0.624i)$
$L(3)$  $\approx$  $0.127879 - 0.364902i$
$L(\frac12)$  $\approx$  $0.127879 - 0.364902i$
$L(\frac{7}{2})$   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 \( 1 + (0.728 - 15.5i)T \)
good2 \( 1 + (7.30 - 2.65i)T + (24.5 - 20.5i)T^{2} \)
5 \( 1 + (-15.2 - 86.2i)T + (-2.93e3 + 1.06e3i)T^{2} \)
7 \( 1 + (90.4 + 75.9i)T + (2.91e3 + 1.65e4i)T^{2} \)
11 \( 1 + (-19.6 + 111. i)T + (-1.51e5 - 5.50e4i)T^{2} \)
13 \( 1 + (178. + 64.9i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (-522. - 905. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.15e3 + 2.00e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (3.29e3 - 2.76e3i)T + (1.11e6 - 6.33e6i)T^{2} \)
29 \( 1 + (4.18e3 - 1.52e3i)T + (1.57e7 - 1.31e7i)T^{2} \)
31 \( 1 + (1.59e3 - 1.33e3i)T + (4.97e6 - 2.81e7i)T^{2} \)
37 \( 1 + (-5.11e3 - 8.85e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (1.01e4 + 3.68e3i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 + (2.35e3 - 1.33e4i)T + (-1.38e8 - 5.02e7i)T^{2} \)
47 \( 1 + (-2.15e4 - 1.80e4i)T + (3.98e7 + 2.25e8i)T^{2} \)
53 \( 1 + 2.81e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.37e3 - 7.78e3i)T + (-6.71e8 + 2.44e8i)T^{2} \)
61 \( 1 + (8.17e3 + 6.86e3i)T + (1.46e8 + 8.31e8i)T^{2} \)
67 \( 1 + (-1.30e3 - 476. i)T + (1.03e9 + 8.67e8i)T^{2} \)
71 \( 1 + (1.22e4 + 2.11e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (8.11e3 - 1.40e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-5.31e4 + 1.93e4i)T + (2.35e9 - 1.97e9i)T^{2} \)
83 \( 1 + (-3.51e4 + 1.28e4i)T + (3.01e9 - 2.53e9i)T^{2} \)
89 \( 1 + (2.70e4 - 4.67e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (1.67e4 - 9.49e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−17.10766083655624219735839051435, −15.99751964692371925641216275259, −15.03471466100570345127495471184, −13.71845452105807548774411134787, −11.14031665323032307099107176244, −10.19273666815753088574154601914, −9.452892654035346003138353211616, −7.60377116192500044324130150292, −6.27527030499717305277960677088, −3.41475036445390225449663172636, 0.37820466802563414420413706963, 1.93779478431077064570146715841, 5.60254865184123420256279651292, 7.74658726451378483740977194296, 8.879901245573587199931082883752, 9.822676839529561708417829696814, 11.91610040276871881069341141210, 12.54784304161893972808604181291, 13.98957510716785513224862834296, 16.32759288904680143271230119359

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