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Properties

Label 2-3e6-9.4-c1-0-28
Degree $2$
Conductor $729$
Sign $-0.500 + 0.866i$
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  + (−0.0864 − 0.149i)2-s + (0.985 − 1.70i)4-s + (1.86 − 3.23i)5-s + (−1.51 − 2.62i)7-s − 0.686·8-s − 0.646·10-s + (1.24 + 2.15i)11-s + (0.382 − 0.662i)13-s + (−0.262 + 0.454i)14-s + (−1.91 − 3.30i)16-s + 4.62·17-s − 0.611·19-s + (−3.68 − 6.37i)20-s + (0.215 − 0.373i)22-s + (−3.26 + 5.65i)23-s + ⋯
L(s)  = 1  + (−0.0611 − 0.105i)2-s + (0.492 − 0.853i)4-s + (0.835 − 1.44i)5-s + (−0.572 − 0.992i)7-s − 0.242·8-s − 0.204·10-s + (0.375 + 0.650i)11-s + (0.106 − 0.183i)13-s + (−0.0700 + 0.121i)14-s + (−0.477 − 0.827i)16-s + 1.12·17-s − 0.140·19-s + (−0.823 − 1.42i)20-s + (0.0459 − 0.0795i)22-s + (−0.680 + 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882073 - 1.52779i\)
\(L(\frac12)\) \(\approx\) \(0.882073 - 1.52779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0864 + 0.149i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.86 + 3.23i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.51 + 2.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.24 - 2.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.382 + 0.662i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.62T + 17T^{2} \)
19 \( 1 + 0.611T + 19T^{2} \)
23 \( 1 + (3.26 - 5.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.27 - 5.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.27 - 5.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.95T + 37T^{2} \)
41 \( 1 + (2.63 - 4.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.78 + 4.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.553 + 0.959i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + (-5.92 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.09 + 7.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.606 + 1.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.91T + 71T^{2} \)
73 \( 1 - 4.29T + 73T^{2} \)
79 \( 1 + (-5.89 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.50 + 7.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.53T + 89T^{2} \)
97 \( 1 + (-0.474 - 0.821i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.849778887314237430323414115794, −9.655895967046183926004748314230, −8.578544990141122950620827185796, −7.38313321482062038725649296489, −6.49326291126725986202412432073, −5.53087849534545741345384266974, −4.87779745470513943072470737667, −3.55273454906528791511073743874, −1.79351611327971733277131532124, −0.967272969664452184742312483763, 2.31203922840097218731575869544, 2.87362636714463341065238374735, 3.88992057776074333978046307748, 5.94420611261581653739615773665, 6.12127444988488684115016865512, 7.06975493277365310941344681912, 8.038502257320591989183560600634, 8.971742566463710308051226357943, 9.889663377561520303886286753164, 10.62242596326873910728596521555

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