Properties

Label 2-3e3-27.22-c3-0-7
Degree $2$
Conductor $27$
Sign $0.320 + 0.947i$
Analytic cond. $1.59305$
Root an. cond. $1.26216$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 − 2.04i)2-s + (−2.73 − 4.41i)3-s + (0.375 − 2.12i)4-s + (6.73 − 2.45i)5-s + (−15.7 − 5.17i)6-s + (0.843 + 4.78i)7-s + (9.30 + 16.1i)8-s + (−12.0 + 24.1i)9-s + (11.4 − 19.7i)10-s + (20.1 + 7.34i)11-s + (−10.4 + 4.16i)12-s + (−26.7 − 22.4i)13-s + (11.8 + 9.95i)14-s + (−29.2 − 23.0i)15-s + (71.9 + 26.2i)16-s + (−57.1 + 99.0i)17-s + ⋯
L(s)  = 1  + (0.863 − 0.724i)2-s + (−0.526 − 0.849i)3-s + (0.0469 − 0.266i)4-s + (0.602 − 0.219i)5-s + (−1.07 − 0.352i)6-s + (0.0455 + 0.258i)7-s + (0.411 + 0.712i)8-s + (−0.444 + 0.895i)9-s + (0.361 − 0.625i)10-s + (0.553 + 0.201i)11-s + (−0.250 + 0.100i)12-s + (−0.570 − 0.478i)13-s + (0.226 + 0.189i)14-s + (−0.503 − 0.396i)15-s + (1.12 + 0.409i)16-s + (−0.815 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(1.59305\)
Root analytic conductor: \(1.26216\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28367 - 0.920568i\)
\(L(\frac12)\) \(\approx\) \(1.28367 - 0.920568i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.73 + 4.41i)T \)
good2 \( 1 + (-2.44 + 2.04i)T + (1.38 - 7.87i)T^{2} \)
5 \( 1 + (-6.73 + 2.45i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-0.843 - 4.78i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (-20.1 - 7.34i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (26.7 + 22.4i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (57.1 - 99.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (75.8 + 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-16.1 + 91.4i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-144. + 121. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (1.10 - 6.29i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (196. - 340. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-159. - 134. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (142. + 51.9i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (29.2 + 165. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 2.81T + 1.48e5T^{2} \)
59 \( 1 + (-231. + 84.1i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (123. + 702. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-181. - 152. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (61.2 - 106. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (137. + 238. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-197. + 166. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-1.01e3 + 855. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (41.5 + 72.0i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-34.8 - 12.6i)T + (6.99e5 + 5.86e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.23684791760929743827253611519, −15.01048769650089866625665554054, −13.54319607961033676603123023863, −12.86274679758233798862561057509, −11.84665225878326324646365945410, −10.61840259167332728436997269453, −8.443022470730587200712291511970, −6.44671050287568610252670457485, −4.81496866556960304674423532184, −2.20208299318965194795403470130, 4.17234169222354352345133373495, 5.59974459069828088722651486801, 6.83075424556446502633622633310, 9.390485940742395624203294413474, 10.55424390650720007932453770225, 12.15530951953707669071916275321, 13.88579575313490249107121612562, 14.54945856616171914915631493625, 15.84249364163678431938654153621, 16.70238838839537369405565232331

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