Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.939 + 0.342i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−3 − 1.73i)5-s + (−1 − 1.73i)7-s + 8.66i·8-s − 6·10-s + (1.5 − 0.866i)11-s + (2 − 3.46i)13-s + (−3 − 1.73i)14-s + (5.5 + 9.52i)16-s − 15.5i·17-s + 11·19-s + (2.99 − 1.73i)20-s + (1.5 − 2.59i)22-s + (24 + 13.8i)23-s + ⋯
L(s)  = 1  + (0.750 − 0.433i)2-s + (−0.125 + 0.216i)4-s + (−0.600 − 0.346i)5-s + (−0.142 − 0.247i)7-s + 1.08i·8-s − 0.600·10-s + (0.136 − 0.0787i)11-s + (0.153 − 0.266i)13-s + (−0.214 − 0.123i)14-s + (0.343 + 0.595i)16-s − 0.916i·17-s + 0.578·19-s + (0.149 − 0.0866i)20-s + (0.0681 − 0.118i)22-s + (1.04 + 0.602i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.939 + 0.342i$
motivic weight  =  \(2\)
character  :  $\chi_{27} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :1),\ 0.939 + 0.342i)$
$L(\frac{3}{2})$  $\approx$  $1.15277 - 0.203265i$
$L(\frac12)$  $\approx$  $1.15277 - 0.203265i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
good2 \( 1 + (-1.5 + 0.866i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (3 + 1.73i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + (-24 - 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (39 - 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16 - 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 34T + 1.36e3T^{2} \)
41 \( 1 + (-10.5 - 6.06i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 24.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (43.5 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (28 + 48.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (19 + 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-42 + 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 - 99.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−16.97852006073033270567206013185, −15.81194033586924826395631381677, −14.32332847564253113049640967588, −13.20879476950558931481134188137, −12.14421051556168363781619320952, −11.06277888648967116387068588430, −9.045968939723155070863090308782, −7.51335180285504062301562843657, −5.10449220145022378540999645957, −3.47851947562111442583082863489, 3.93770391264367289528376419738, 5.76332489329024944103761021810, 7.29002998337248567308671618499, 9.251625512923039693916806155580, 10.88481014932351468344588166574, 12.41975491966578491170170802382, 13.63945946502705838048830076409, 14.90252909148791652513749596453, 15.54281834585365900676257331795, 16.96053097621608547095111719016

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