Properties

Degree 2
Conductor $ 3^{3} $
Sign $-0.742 + 0.669i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.982 − 0.357i)2-s + (−5.19 − 0.0395i)3-s + (−5.29 − 4.43i)4-s + (2.68 − 15.1i)5-s + (5.09 + 1.89i)6-s + (−18.0 + 15.1i)7-s + (7.79 + 13.5i)8-s + (26.9 + 0.411i)9-s + (−8.07 + 13.9i)10-s + (−7.09 − 40.2i)11-s + (27.3 + 23.2i)12-s + (11.9 − 4.34i)13-s + (23.1 − 8.42i)14-s + (−14.5 + 78.8i)15-s + (6.76 + 38.3i)16-s + (23.8 − 41.2i)17-s + ⋯
L(s)  = 1  + (−0.347 − 0.126i)2-s + (−0.999 − 0.00761i)3-s + (−0.661 − 0.554i)4-s + (0.239 − 1.35i)5-s + (0.346 + 0.129i)6-s + (−0.973 + 0.817i)7-s + (0.344 + 0.596i)8-s + (0.999 + 0.0152i)9-s + (−0.255 + 0.442i)10-s + (−0.194 − 1.10i)11-s + (0.657 + 0.559i)12-s + (0.254 − 0.0926i)13-s + (0.441 − 0.160i)14-s + (−0.250 + 1.35i)15-s + (0.105 + 0.599i)16-s + (0.339 − 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-0.742 + 0.669i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ -0.742 + 0.669i)$
$L(2)$  $\approx$  $0.161886 - 0.421456i$
$L(\frac12)$  $\approx$  $0.161886 - 0.421456i$
$L(\frac{5}{2})$   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 \( 1 + (5.19 + 0.0395i)T \)
good2 \( 1 + (0.982 + 0.357i)T + (6.12 + 5.14i)T^{2} \)
5 \( 1 + (-2.68 + 15.1i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (18.0 - 15.1i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (7.09 + 40.2i)T + (-1.25e3 + 455. i)T^{2} \)
13 \( 1 + (-11.9 + 4.34i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (-23.8 + 41.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (67.6 + 117. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36.2 - 30.4i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (118. + 43.2i)T + (1.86e4 + 1.56e4i)T^{2} \)
31 \( 1 + (-158. - 132. i)T + (5.17e3 + 2.93e4i)T^{2} \)
37 \( 1 + (-93.6 + 162. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (41.7 - 15.2i)T + (5.27e4 - 4.43e4i)T^{2} \)
43 \( 1 + (-41.2 - 234. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (-83.8 + 70.3i)T + (1.80e4 - 1.02e5i)T^{2} \)
53 \( 1 - 11.6T + 1.48e5T^{2} \)
59 \( 1 + (-126. + 715. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (45.5 - 38.2i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (290. - 105. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-130. + 226. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-186. - 323. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (128. + 46.9i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (-1.03e3 - 376. i)T + (4.38e5 + 3.67e5i)T^{2} \)
89 \( 1 + (716. + 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-137. - 780. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−16.52826453000411116669349780229, −15.61381981849628426443429281689, −13.44163055549524274831034739254, −12.69031803200426040689283584116, −11.17044326003117114345156221392, −9.610600992045367482335563499888, −8.729086011840217425227170297172, −6.01332145356592420027285843104, −4.96382607162700601833198861256, −0.57748247998975939620741274177, 3.94417962578661271917255253958, 6.40305121736954872508937963713, 7.46951790572356534145568221439, 9.918720550596395109061919555930, 10.49264699395342300347403903349, 12.35447376669603795956783313097, 13.40431806466835301868225041324, 14.98077671950524233024347306635, 16.55811163766091200376595077354, 17.27801003991533407114968610624

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