Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.814 - 0.580i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.904 + 5.13i)2-s + (−4.65 − 2.30i)3-s + (−17.9 + 6.54i)4-s + (10.7 + 9.02i)5-s + (7.58 − 25.9i)6-s + (9.67 + 3.52i)7-s + (−29.0 − 50.2i)8-s + (16.4 + 21.4i)9-s + (−36.5 + 63.3i)10-s + (21.5 − 18.0i)11-s + (98.8 + 10.8i)12-s + (−1.35 + 7.65i)13-s + (−9.31 + 52.8i)14-s + (−29.3 − 66.8i)15-s + (114. − 96.0i)16-s + (4.59 − 7.96i)17-s + ⋯
L(s)  = 1  + (0.319 + 1.81i)2-s + (−0.896 − 0.442i)3-s + (−2.24 + 0.818i)4-s + (0.962 + 0.807i)5-s + (0.516 − 1.76i)6-s + (0.522 + 0.190i)7-s + (−1.28 − 2.22i)8-s + (0.608 + 0.793i)9-s + (−1.15 + 2.00i)10-s + (0.590 − 0.495i)11-s + (2.37 + 0.261i)12-s + (−0.0288 + 0.163i)13-s + (−0.177 + 1.00i)14-s + (−0.505 − 1.15i)15-s + (1.78 − 1.50i)16-s + (0.0656 − 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.814 - 0.580i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.814 - 0.580i)$
$L(2)$  $\approx$  $0.335656 + 1.04911i$
$L(\frac12)$  $\approx$  $0.335656 + 1.04911i$
$L(\frac{5}{2})$   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 \( 1 + (4.65 + 2.30i)T \)
good2 \( 1 + (-0.904 - 5.13i)T + (-7.51 + 2.73i)T^{2} \)
5 \( 1 + (-10.7 - 9.02i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-9.67 - 3.52i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-21.5 + 18.0i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (1.35 - 7.65i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-4.59 + 7.96i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46.7 + 81.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-144. + 52.6i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-30.3 - 171. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (199. - 72.4i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-51.0 + 88.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-7.73 + 43.8i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (10.2 - 8.57i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-329. - 119. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + 102.T + 1.48e5T^{2} \)
59 \( 1 + (484. + 406. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-549. - 199. i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-128. + 726. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (476. - 826. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (489. + 847. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-30.7 - 174. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (141. + 800. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (-402. - 696. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (274. - 230. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−17.27341549214488103968427783519, −16.29542217053022345065335217868, −14.84019121724066816521232609459, −13.98513796373329365298581618876, −12.85364569996612651044058996314, −10.92598175022010902503823056019, −8.956646963090505343742140192796, −7.14107823346356782242475015457, −6.28919678628565855613009312141, −5.05637231458117203118584039595, 1.42069217992721931366796728410, 4.27783707078633295668451362192, 5.53482035366698222405761473507, 9.177090462673890726722430470513, 10.09665642139307057333902907681, 11.25474190905946014758395497398, 12.36987881380347513972207307449, 13.27052911466819132501197089888, 14.73098064843465713350631527145, 16.96966368264735697247969934962

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