Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.822 - 0.568i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0518 − 0.294i)2-s + (−2.82 + 4.36i)3-s + (7.43 − 2.70i)4-s + (15.3 + 12.8i)5-s + (1.43 + 0.603i)6-s + (−20.1 − 7.35i)7-s + (−2.37 − 4.11i)8-s + (−11.0 − 24.6i)9-s + (2.99 − 5.18i)10-s + (0.558 − 0.468i)11-s + (−9.16 + 40.0i)12-s + (5.34 − 30.3i)13-s + (−1.11 + 6.32i)14-s + (−99.5 + 30.6i)15-s + (47.3 − 39.7i)16-s + (0.666 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.0183 − 0.104i)2-s + (−0.542 + 0.839i)3-s + (0.929 − 0.338i)4-s + (1.37 + 1.15i)5-s + (0.0973 + 0.0410i)6-s + (−1.09 − 0.396i)7-s + (−0.105 − 0.181i)8-s + (−0.410 − 0.911i)9-s + (0.0947 − 0.164i)10-s + (0.0153 − 0.0128i)11-s + (−0.220 + 0.963i)12-s + (0.114 − 0.647i)13-s + (−0.0212 + 0.120i)14-s + (−1.71 + 0.527i)15-s + (0.740 − 0.621i)16-s + (0.00950 − 0.0164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.822 - 0.568i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.822 - 0.568i)$
$L(2)$  $\approx$  $1.20217 + 0.375333i$
$L(\frac12)$  $\approx$  $1.20217 + 0.375333i$
$L(\frac{5}{2})$   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 \( 1 + (2.82 - 4.36i)T \)
good2 \( 1 + (0.0518 + 0.294i)T + (-7.51 + 2.73i)T^{2} \)
5 \( 1 + (-15.3 - 12.8i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (20.1 + 7.35i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-0.558 + 0.468i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (-5.34 + 30.3i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-0.666 + 1.15i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (34.7 + 60.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (110. - 40.3i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (15.9 + 90.6i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-175. + 63.9i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (74.3 - 128. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (55.1 - 312. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (220. - 185. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-19.8 - 7.23i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 - 136.T + 1.48e5T^{2} \)
59 \( 1 + (-36.7 - 30.8i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-509. - 185. i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (15.9 - 90.5i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (531. - 920. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (342. + 592. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (116. + 659. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-96.8 - 549. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (269. + 466. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-226. + 190. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−16.92141340114298880326237718826, −15.72772480476267001881162940949, −14.75790127444140913032723153812, −13.33042303615082367658142746243, −11.44826271323311411245825690853, −10.20299929542336364172814438391, −9.893088793606037974512438453048, −6.67349801434073957556847376316, −5.92086431659915349002172281956, −2.96703175399403924478302794791, 1.98787838818610936603492684719, 5.73187938691856739965772187660, 6.62636341609554583680753947386, 8.601844275616317860940043446502, 10.20754083782924674973280321704, 12.10290120936457886268237462859, 12.72214730256413832673112925852, 13.89005832569254031242408340715, 16.10273724997725069229138583911, 16.70050032473511060017476033899

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