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  :  $\chi_{27} (25, \cdot )$
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

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

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