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Properties

Label 2-3e6-9.7-c1-0-1
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: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.500 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389061 + 0.673875i\)
\(L(\frac12)\) \(\approx\) \(0.389061 + 0.673875i\)
\(L(\frac{3}{2})\) 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

−11.07076021668948645169545005284, −9.365540650670453256477851306169, −9.063554679754861369093948665336, −8.071181574314172379531402590266, −7.44495892294792785942431371486, −6.29744007659574312309626431420, −5.08652586481928254245805698958, −4.26167788645797793677730789379, −3.17195365667453867357634180174, −1.87330447859963838101780506244, 0.36808030650354557377827098369, 2.45236387474715049476445580038, 3.43499151416229099517119035611, 4.48872805422083980398877964233, 5.94058423410793550206935583879, 6.76088714147951488576534681720, 7.14534917594546971262687262458, 8.166973954349805366076269801237, 9.477958266585090248851411551734, 10.52101974596348809731863288497

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