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Properties

Label 2-3e6-27.7-c1-0-13
Degree $2$
Conductor $729$
Sign $0.973 + 0.230i$
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  + (−1.53 + 1.28i)4-s + (−3.06 − 2.57i)7-s + (6.57 + 2.39i)13-s + (0.694 − 3.93i)16-s + (0.5 − 0.866i)19-s + (4.69 − 1.71i)25-s + 8·28-s + (8.42 − 7.07i)31-s + (5 + 8.66i)37-s + (0.868 − 4.92i)43-s + (1.56 + 8.86i)49-s + (−13.1 + 4.78i)52-s + (−0.766 − 0.642i)61-s + (4.00 + 6.92i)64-s + (−4.69 − 1.71i)67-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)4-s + (−1.15 − 0.971i)7-s + (1.82 + 0.664i)13-s + (0.173 − 0.984i)16-s + (0.114 − 0.198i)19-s + (0.939 − 0.342i)25-s + 1.51·28-s + (1.51 − 1.26i)31-s + (0.821 + 1.42i)37-s + (0.132 − 0.750i)43-s + (0.223 + 1.26i)49-s + (−1.82 + 0.664i)52-s + (−0.0980 − 0.0823i)61-s + (0.500 + 0.866i)64-s + (−0.574 − 0.208i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13653 - 0.132842i\)
\(L(\frac12)\) \(\approx\) \(1.13653 - 0.132842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.06 + 2.57i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-6.57 - 2.39i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-8.42 + 7.07i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.868 + 4.92i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (0.766 + 0.642i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.69 + 1.71i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.2 + 4.44i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.868 + 4.92i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.23214464087526367844708396379, −9.472377073703819482909628472325, −8.678329946907627334241885756574, −7.88360060745879125646391381049, −6.77197118708064399064719947821, −6.14643423188643526032020318698, −4.58885160991286545714637689621, −3.85779063245516187222085309511, −3.02029054062207989109891076167, −0.823563746339314533382540274738, 1.05832123159129290306861063711, 2.88971687724150994353734800156, 3.86852740133365889913321318730, 5.18741081629584981848113567264, 5.99499604486112861398323860844, 6.54964488385060639079402672367, 8.152597809530267041362975953677, 8.860207680636327589962838723666, 9.446314362089780396250193881313, 10.37719194985201615781198833003

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