Properties

Label 2-3e3-27.4-c7-0-2
Degree $2$
Conductor $27$
Sign $-0.995 - 0.0934i$
Analytic cond. $8.43439$
Root an. cond. $2.90420$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.29 + 1.56i)2-s + (11.1 + 45.4i)3-s + (−82.0 − 68.8i)4-s + (−20.3 + 115. i)5-s + (−23.2 + 212. i)6-s + (−610. + 512. i)7-s + (−537. − 931. i)8-s + (−1.93e3 + 1.01e3i)9-s + (−267. + 464. i)10-s + (−574. − 3.25e3i)11-s + (2.21e3 − 4.49e3i)12-s + (−6.69e3 + 2.43e3i)13-s + (−3.42e3 + 1.24e3i)14-s + (−5.46e3 + 359. i)15-s + (1.52e3 + 8.65e3i)16-s + (−1.16e4 + 2.01e4i)17-s + ⋯
L(s)  = 1  + (0.379 + 0.138i)2-s + (0.237 + 0.971i)3-s + (−0.640 − 0.537i)4-s + (−0.0727 + 0.412i)5-s + (−0.0438 + 0.401i)6-s + (−0.672 + 0.564i)7-s + (−0.371 − 0.643i)8-s + (−0.886 + 0.462i)9-s + (−0.0847 + 0.146i)10-s + (−0.130 − 0.738i)11-s + (0.369 − 0.750i)12-s + (−0.845 + 0.307i)13-s + (−0.333 + 0.121i)14-s + (−0.418 + 0.0275i)15-s + (0.0931 + 0.528i)16-s + (−0.573 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0934i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.995 - 0.0934i$
Analytic conductor: \(8.43439\)
Root analytic conductor: \(2.90420\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :7/2),\ -0.995 - 0.0934i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0355008 + 0.757802i\)
\(L(\frac12)\) \(\approx\) \(0.0355008 + 0.757802i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.1 - 45.4i)T \)
good2 \( 1 + (-4.29 - 1.56i)T + (98.0 + 82.2i)T^{2} \)
5 \( 1 + (20.3 - 115. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (610. - 512. i)T + (1.43e5 - 8.11e5i)T^{2} \)
11 \( 1 + (574. + 3.25e3i)T + (-1.83e7 + 6.66e6i)T^{2} \)
13 \( 1 + (6.69e3 - 2.43e3i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (1.16e4 - 2.01e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (4.58e3 + 7.94e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-3.31e4 - 2.78e4i)T + (5.91e8 + 3.35e9i)T^{2} \)
29 \( 1 + (-1.08e4 - 3.95e3i)T + (1.32e10 + 1.10e10i)T^{2} \)
31 \( 1 + (-1.91e5 - 1.61e5i)T + (4.77e9 + 2.70e10i)T^{2} \)
37 \( 1 + (-1.68e5 + 2.92e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (2.25e5 - 8.20e4i)T + (1.49e11 - 1.25e11i)T^{2} \)
43 \( 1 + (1.45e4 + 8.27e4i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (8.67e5 - 7.27e5i)T + (8.79e10 - 4.98e11i)T^{2} \)
53 \( 1 + 1.76e6T + 1.17e12T^{2} \)
59 \( 1 + (-2.64e4 + 1.50e5i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (-1.52e6 + 1.27e6i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (3.19e6 - 1.16e6i)T + (4.64e12 - 3.89e12i)T^{2} \)
71 \( 1 + (-1.63e6 + 2.84e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (-1.48e6 - 2.57e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (3.46e6 + 1.26e6i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (1.47e6 + 5.38e5i)T + (2.07e13 + 1.74e13i)T^{2} \)
89 \( 1 + (2.24e5 + 3.88e5i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (1.73e6 + 9.82e6i)T + (-7.59e13 + 2.76e13i)T^{2} \)
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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

−15.97090855527624265015320338936, −15.06189868781226598021006084007, −14.17681515549861501896675698659, −12.84150881374764401936719487901, −10.97961102437251137388121718086, −9.753914922112186594682380776479, −8.716865994349490756441533215007, −6.24008553605587897607517991163, −4.79699049775584831131445920737, −3.16092784529359048884127624545, 0.32156606090034375980606725319, 2.81411454407443043699351921434, 4.74135701838154376455396358374, 6.89547550549589586497462634429, 8.215665771997350745582425841530, 9.643554306948024279754083070989, 11.86401018615092651593378041959, 12.83120780302007216923194887020, 13.51515756335678028967916880854, 14.80754677678802077490750655432

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