Properties

Label 2-3e3-27.4-c5-0-2
Degree $2$
Conductor $27$
Sign $0.888 - 0.459i$
Analytic cond. $4.33036$
Root an. cond. $2.08095$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.55 − 2.02i)2-s + (−15.5 − 1.44i)3-s + (2.28 + 1.91i)4-s + (−3.90 + 22.1i)5-s + (83.3 + 39.4i)6-s + (69.5 − 58.3i)7-s + (85.8 + 148. i)8-s + (238. + 45.0i)9-s + (66.5 − 115. i)10-s + (47.2 + 268. i)11-s + (−32.6 − 33.0i)12-s + (263. − 95.7i)13-s + (−504. + 183. i)14-s + (92.7 − 338. i)15-s + (−192. − 1.09e3i)16-s + (−736. + 1.27e3i)17-s + ⋯
L(s)  = 1  + (−0.982 − 0.357i)2-s + (−0.995 − 0.0930i)3-s + (0.0713 + 0.0599i)4-s + (−0.0698 + 0.396i)5-s + (0.944 + 0.447i)6-s + (0.536 − 0.450i)7-s + (0.474 + 0.821i)8-s + (0.982 + 0.185i)9-s + (0.210 − 0.364i)10-s + (0.117 + 0.668i)11-s + (−0.0655 − 0.0662i)12-s + (0.431 − 0.157i)13-s + (−0.688 + 0.250i)14-s + (0.106 − 0.388i)15-s + (−0.188 − 1.06i)16-s + (−0.618 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(4.33036\)
Root analytic conductor: \(2.08095\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :5/2),\ 0.888 - 0.459i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.557580 + 0.135526i\)
\(L(\frac12)\) \(\approx\) \(0.557580 + 0.135526i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (15.5 + 1.44i)T \)
good2 \( 1 + (5.55 + 2.02i)T + (24.5 + 20.5i)T^{2} \)
5 \( 1 + (3.90 - 22.1i)T + (-2.93e3 - 1.06e3i)T^{2} \)
7 \( 1 + (-69.5 + 58.3i)T + (2.91e3 - 1.65e4i)T^{2} \)
11 \( 1 + (-47.2 - 268. i)T + (-1.51e5 + 5.50e4i)T^{2} \)
13 \( 1 + (-263. + 95.7i)T + (2.84e5 - 2.38e5i)T^{2} \)
17 \( 1 + (736. - 1.27e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-121. - 209. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (101. + 85.4i)T + (1.11e6 + 6.33e6i)T^{2} \)
29 \( 1 + (-7.78e3 - 2.83e3i)T + (1.57e7 + 1.31e7i)T^{2} \)
31 \( 1 + (-5.83e3 - 4.89e3i)T + (4.97e6 + 2.81e7i)T^{2} \)
37 \( 1 + (5.73e3 - 9.92e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (5.44e3 - 1.98e3i)T + (8.87e7 - 7.44e7i)T^{2} \)
43 \( 1 + (3.07e3 + 1.74e4i)T + (-1.38e8 + 5.02e7i)T^{2} \)
47 \( 1 + (-9.83e3 + 8.24e3i)T + (3.98e7 - 2.25e8i)T^{2} \)
53 \( 1 + 1.98e4T + 4.18e8T^{2} \)
59 \( 1 + (2.93e3 - 1.66e4i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (4.75e3 - 3.98e3i)T + (1.46e8 - 8.31e8i)T^{2} \)
67 \( 1 + (2.15e4 - 7.84e3i)T + (1.03e9 - 8.67e8i)T^{2} \)
71 \( 1 + (-2.24e4 + 3.88e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-3.20e4 - 5.55e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-9.52e4 - 3.46e4i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (9.48e3 + 3.45e3i)T + (3.01e9 + 2.53e9i)T^{2} \)
89 \( 1 + (2.29e4 + 3.96e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (2.42e4 + 1.37e5i)T + (-8.06e9 + 2.93e9i)T^{2} \)
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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

−16.98763987484400249471401678125, −15.40055860665178854023288730023, −13.85607967989031062023866917352, −12.16931252957469955797903648426, −10.81488554161802496768981907525, −10.25057484999014800257422148404, −8.406738012261410069766741556932, −6.79104782620470872686238839682, −4.78617488371217495533044063000, −1.32664773378442225718912158758, 0.71156221206185277815590050833, 4.68346689277667383262886799418, 6.51957535466527886693315979550, 8.191716664363491678290182906709, 9.417637459931250507006104809836, 10.92193532399989376525008062895, 12.12664083791562871647151078396, 13.59821267091898388503268751871, 15.68230868537600477280920605629, 16.37265696448940783544821362494

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