Properties

Label 2-3e3-27.13-c3-0-5
Degree $2$
Conductor $27$
Sign $0.849 + 0.528i$
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  + (0.111 + 0.634i)2-s + (−0.446 − 5.17i)3-s + (7.12 − 2.59i)4-s + (−0.393 − 0.330i)5-s + (3.23 − 0.862i)6-s + (9.02 + 3.28i)7-s + (5.02 + 8.69i)8-s + (−26.6 + 4.62i)9-s + (0.165 − 0.287i)10-s + (−31.1 + 26.1i)11-s + (−16.6 − 35.7i)12-s + (−0.304 + 1.72i)13-s + (−1.07 + 6.09i)14-s + (−1.53 + 2.18i)15-s + (41.5 − 34.8i)16-s + (−37.5 + 65.1i)17-s + ⋯
L(s)  = 1  + (0.0395 + 0.224i)2-s + (−0.0859 − 0.996i)3-s + (0.890 − 0.324i)4-s + (−0.0352 − 0.0295i)5-s + (0.220 − 0.0587i)6-s + (0.487 + 0.177i)7-s + (0.221 + 0.384i)8-s + (−0.985 + 0.171i)9-s + (0.00524 − 0.00907i)10-s + (−0.853 + 0.716i)11-s + (−0.399 − 0.859i)12-s + (−0.00649 + 0.0368i)13-s + (−0.0205 + 0.116i)14-s + (−0.0264 + 0.0376i)15-s + (0.648 − 0.544i)16-s + (−0.536 + 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.849 + 0.528i$
Analytic conductor: \(1.59305\)
Root analytic conductor: \(1.26216\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :3/2),\ 0.849 + 0.528i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.27876 - 0.365202i\)
\(L(\frac12)\) \(\approx\) \(1.27876 - 0.365202i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.446 + 5.17i)T \)
good2 \( 1 + (-0.111 - 0.634i)T + (-7.51 + 2.73i)T^{2} \)
5 \( 1 + (0.393 + 0.330i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-9.02 - 3.28i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (31.1 - 26.1i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (0.304 - 1.72i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (37.5 - 65.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-42.1 - 73.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-121. + 44.1i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (45.3 + 257. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (284. - 103. i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (103. - 178. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-54.5 + 309. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (-99.6 + 83.6i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (186. + 67.9i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 - 527.T + 1.48e5T^{2} \)
59 \( 1 + (59.2 + 49.6i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-251. - 91.6i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-15.6 + 88.6i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (360. - 624. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-66.7 - 115. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-21.9 - 124. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-146. - 832. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (693. + 1.20e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (216. - 181. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−16.82967379590989156091322296531, −15.41468877053338737685903275087, −14.39045049665587174673391432890, −12.87677340365525254838663578896, −11.74400483949329772795042915685, −10.49992894263189078625673619559, −8.205429935617050613071804736453, −7.04195511122095828166989857045, −5.57477092573657061416331953334, −2.07480092707585363972836243542, 3.14504001774021647626870029415, 5.25658111834266303438356147008, 7.37610840207605996895759657387, 9.135113145889426411485655586986, 10.92132347982990976340358438547, 11.30150583099460447880142734439, 13.17013554700411165874453723104, 14.79435890242326944299879885638, 15.86664618944562335112541757994, 16.62924314926765729276916858198

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