Properties

Degree 2
Conductor $ 3^{3} $
Sign $0.308 - 0.951i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.640 + 3.63i)2-s + (5.12 − 0.863i)3-s + (−5.26 + 1.91i)4-s + (−1.23 − 1.03i)5-s + (6.41 + 18.0i)6-s + (−31.9 − 11.6i)7-s + (4.41 + 7.64i)8-s + (25.5 − 8.84i)9-s + (2.97 − 5.16i)10-s + (18.8 − 15.8i)11-s + (−25.3 + 14.3i)12-s + (5.35 − 30.3i)13-s + (21.7 − 123. i)14-s + (−7.23 − 4.25i)15-s + (−59.3 + 49.7i)16-s + (−18.2 + 31.5i)17-s + ⋯
L(s)  = 1  + (0.226 + 1.28i)2-s + (0.986 − 0.166i)3-s + (−0.658 + 0.239i)4-s + (−0.110 − 0.0928i)5-s + (0.436 + 1.22i)6-s + (−1.72 − 0.627i)7-s + (0.195 + 0.337i)8-s + (0.944 − 0.327i)9-s + (0.0942 − 0.163i)10-s + (0.517 − 0.434i)11-s + (−0.609 + 0.345i)12-s + (0.114 − 0.647i)13-s + (0.415 − 2.35i)14-s + (−0.124 − 0.0731i)15-s + (−0.926 + 0.777i)16-s + (−0.260 + 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.308 - 0.951i$
motivic weight  =  \(3\)
character  :  $\chi_{27} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 27,\ (\ :3/2),\ 0.308 - 0.951i)$
$L(2)$  $\approx$  $1.25569 + 0.912836i$
$L(\frac12)$  $\approx$  $1.25569 + 0.912836i$
$L(\frac{5}{2})$   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 \( 1 + (-5.12 + 0.863i)T \)
good2 \( 1 + (-0.640 - 3.63i)T + (-7.51 + 2.73i)T^{2} \)
5 \( 1 + (1.23 + 1.03i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (31.9 + 11.6i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-18.8 + 15.8i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (-5.35 + 30.3i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (18.2 - 31.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-36.5 - 63.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (62.9 - 22.9i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-41.0 - 232. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (152. - 55.4i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-93.1 + 161. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (37.9 - 215. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (-156. + 131. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (450. + 163. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 - 736.T + 1.48e5T^{2} \)
59 \( 1 + (-39.2 - 32.9i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-108. - 39.5i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-20.2 + 114. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (-118. + 205. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (23.0 + 39.9i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (130. + 741. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (59.3 + 336. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (455. + 789. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (37.6 - 31.6i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−16.46291280749346589789727623104, −16.00078598518588352870286379480, −14.68967904710405194239793848882, −13.69851730188000118879720679191, −12.69888621907996947093605154743, −10.19140927757705028998719418479, −8.714086531502905219873597730103, −7.33809389958850298070033129986, −6.20610285918560181958197032825, −3.64857059629298386751389082339, 2.54649121263162921782917100208, 3.88914895378858045229658181180, 6.91266352917987661501882500921, 9.260090221208959535636632470780, 9.840754751251746726330787714462, 11.63404752304219248057596436336, 12.83359841371406666031282434631, 13.66661303870652203834854606933, 15.34977857912185424363262745176, 16.29533459502214506270615578253

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