Properties

Label 2-3e6-1.1-c1-0-20
Degree $2$
Conductor $729$
Sign $1$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.70·2-s + 5.31·4-s − 1.67·5-s + 0.500·7-s + 8.97·8-s − 4.52·10-s − 1.91·11-s + 3.11·13-s + 1.35·14-s + 13.6·16-s + 2.66·17-s + 5.79·19-s − 8.89·20-s − 5.18·22-s − 4.64·23-s − 2.20·25-s + 8.41·26-s + 2.66·28-s − 2.61·29-s − 4.61·31-s + 18.9·32-s + 7.20·34-s − 0.837·35-s − 4.85·37-s + 15.6·38-s − 15.0·40-s − 11.5·41-s + ⋯
L(s)  = 1  + 1.91·2-s + 2.65·4-s − 0.747·5-s + 0.189·7-s + 3.17·8-s − 1.43·10-s − 0.578·11-s + 0.863·13-s + 0.361·14-s + 3.40·16-s + 0.646·17-s + 1.32·19-s − 1.98·20-s − 1.10·22-s − 0.968·23-s − 0.440·25-s + 1.65·26-s + 0.503·28-s − 0.485·29-s − 0.828·31-s + 3.34·32-s + 1.23·34-s − 0.141·35-s − 0.798·37-s + 2.54·38-s − 2.37·40-s − 1.80·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(4.457404053\)
\(L(\frac12)\) \(\approx\) \(4.457404053\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 2T^{2} \)
5 \( 1 + 1.67T + 5T^{2} \)
7 \( 1 - 0.500T + 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
23 \( 1 + 4.64T + 23T^{2} \)
29 \( 1 + 2.61T + 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 + 4.85T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 9.00T + 43T^{2} \)
47 \( 1 - 6.83T + 47T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 - 6.84T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 2.83T + 71T^{2} \)
73 \( 1 - 9.93T + 73T^{2} \)
79 \( 1 - 5.31T + 79T^{2} \)
83 \( 1 + 2.72T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 6.88T + 97T^{2} \)
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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

−10.83825390502468082769418293347, −9.895134984675393727838785945861, −8.206352075220452793699290250844, −7.56942209708635272007851485562, −6.64584399877314293437244697816, −5.58745861719681841202894793965, −5.01427103632711479399913982634, −3.74249690306772619464437519023, −3.35179486421818005857654391513, −1.81702693168411902282057463504, 1.81702693168411902282057463504, 3.35179486421818005857654391513, 3.74249690306772619464437519023, 5.01427103632711479399913982634, 5.58745861719681841202894793965, 6.64584399877314293437244697816, 7.56942209708635272007851485562, 8.206352075220452793699290250844, 9.895134984675393727838785945861, 10.83825390502468082769418293347

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