Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.611 + 0.791i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.06 − 3.41i)2-s + (−1.93 + 4.82i)3-s + (3.50 − 19.8i)4-s + (−5.18 + 1.88i)5-s + (8.57 + 26.2i)6-s + (4.27 + 24.2i)7-s + (−32.2 − 55.9i)8-s + (−19.5 − 18.6i)9-s + (−14.6 + 25.3i)10-s + (0.554 + 0.201i)11-s + (88.9 + 55.3i)12-s + (−1.25 − 1.05i)13-s + (100. + 84.0i)14-s + (0.937 − 28.6i)15-s + (−170. − 62.0i)16-s + (33.3 − 57.6i)17-s + ⋯
L(s)  = 1  + (1.43 − 1.20i)2-s + (−0.372 + 0.927i)3-s + (0.437 − 2.48i)4-s + (−0.463 + 0.168i)5-s + (0.583 + 1.78i)6-s + (0.230 + 1.30i)7-s + (−1.42 − 2.47i)8-s + (−0.722 − 0.691i)9-s + (−0.462 + 0.801i)10-s + (0.0152 + 0.00553i)11-s + (2.14 + 1.33i)12-s + (−0.0268 − 0.0225i)13-s + (1.91 + 1.60i)14-s + (0.0161 − 0.492i)15-s + (−2.66 − 0.969i)16-s + (0.475 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.611 + 0.791i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.611 + 0.791i)$
$L(2)$  $\approx$  $1.72541 - 0.847006i$
$L(\frac12)$  $\approx$  $1.72541 - 0.847006i$
$L(\frac{5}{2})$   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 \( 1 + (1.93 - 4.82i)T \)
good2 \( 1 + (-4.06 + 3.41i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (5.18 - 1.88i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-4.27 - 24.2i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (-0.554 - 0.201i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (1.25 + 1.05i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-33.3 + 57.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-6.62 - 11.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-28.1 + 159. i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (54.0 - 45.3i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (24.1 - 137. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (40.2 - 69.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-130. - 109. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-456. - 166. i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-17.5 - 99.3i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 374.T + 1.48e5T^{2} \)
59 \( 1 + (97.7 - 35.5i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-85.6 - 485. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (567. + 476. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-96.7 + 167. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (400. + 694. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-899. + 754. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (918. - 770. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (694. + 1.20e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (413. + 150. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−16.06683650205129104195741500569, −15.09297388671678766861216227666, −14.30766726690575310015018551927, −12.47640146571388763731471707337, −11.71400220146776053483979635914, −10.72379408033116987721125106011, −9.283847846780742373364112672033, −5.86871628198039229119774045522, −4.67616862239240488925019471336, −2.98587515367728897416871381677, 4.00908443611332755794777030934, 5.73189330536702766820685614977, 7.21833877167778151728093511558, 7.925357879845483883295042273512, 11.26832743088452748791233411985, 12.49778875203119819448050985011, 13.49246456157568028492606642505, 14.30207592467338234961357726808, 15.72330984035532747065662818145, 16.93663001679087570026890663532

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