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Properties

Label 2-3e6-27.25-c1-0-13
Degree $2$
Conductor $729$
Sign $-0.116 - 0.993i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.118 + 0.673i)2-s + (1.43 + 0.524i)4-s + (−0.802 + 0.673i)5-s + (0.113 − 0.0412i)7-s + (−1.20 + 2.09i)8-s + (−0.358 − 0.620i)10-s + (4.16 + 3.49i)11-s + (−0.794 − 4.50i)13-s + (0.0143 + 0.0812i)14-s + (1.08 + 0.907i)16-s + (2.38 + 4.13i)17-s + (0.294 − 0.509i)19-s + (−1.50 + 0.549i)20-s + (−2.84 + 2.38i)22-s + (−7.32 − 2.66i)23-s + ⋯
L(s)  = 1  + (−0.0839 + 0.476i)2-s + (0.719 + 0.262i)4-s + (−0.359 + 0.301i)5-s + (0.0428 − 0.0155i)7-s + (−0.427 + 0.739i)8-s + (−0.113 − 0.196i)10-s + (1.25 + 1.05i)11-s + (−0.220 − 1.24i)13-s + (0.00382 + 0.0217i)14-s + (0.270 + 0.226i)16-s + (0.579 + 1.00i)17-s + (0.0675 − 0.116i)19-s + (−0.337 + 0.122i)20-s + (−0.607 + 0.509i)22-s + (−1.52 − 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.116 - 0.993i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10340 + 1.23988i\)
\(L(\frac12)\) \(\approx\) \(1.10340 + 1.23988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.118 - 0.673i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.802 - 0.673i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.113 + 0.0412i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-4.16 - 3.49i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.794 + 4.50i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.38 - 4.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.294 + 0.509i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.32 + 2.66i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.880 - 4.99i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-8.23 - 2.99i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.09 - 1.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 7.44i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1 + 0.839i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.27 + 0.826i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 + (-0.0336 + 0.0282i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-9.59 + 3.49i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.322 + 1.83i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (3.25 + 5.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.11 - 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.121 + 0.691i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.17 - 6.67i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-3.42 + 5.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.91 + 5.80i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.52558525683564270878820393590, −9.921567360978443820498486200009, −8.625318137016238253904861917678, −7.915950702661919477052707389532, −7.13950886958184406815143406961, −6.39853653814362893390645565891, −5.46792663807120536977249553499, −4.09059409237896830166057466752, −3.09258351734180223961451967150, −1.71993521062899748954086536967, 0.927255857591062732309210297914, 2.24214692403022444161756114451, 3.53668750172022430466834951789, 4.42964974863425480410509938417, 5.95397823547172842588345583898, 6.49184240988315502854833727361, 7.57205587879261426109483584749, 8.520886322075752522206995689993, 9.574051063570426087683391035992, 10.01101781390596414747386271991

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