↑

Properties

Label 2-3e6-27.5-c0-0-0
Degree $2$
Conductor $729$
Sign $0.230 + 0.973i$
Analytic cond. $0.363818$
Root an. cond. $0.603173$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)7-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)19-s + (0.766 − 0.642i)25-s − 0.999·28-s + (−0.173 + 0.984i)31-s + (0.5 − 0.866i)37-s + (0.939 + 0.342i)43-s + (−0.766 + 0.642i)52-s + (0.347 + 1.96i)61-s + (−0.5 + 0.866i)64-s + (1.53 + 1.28i)67-s + (−1 − 1.73i)73-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)7-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)16-s + (0.5 + 0.866i)19-s + (0.766 − 0.642i)25-s − 0.999·28-s + (−0.173 + 0.984i)31-s + (0.5 − 0.866i)37-s + (0.939 + 0.342i)43-s + (−0.766 + 0.642i)52-s + (0.347 + 1.96i)61-s + (−0.5 + 0.866i)64-s + (1.53 + 1.28i)67-s + (−1 − 1.73i)73-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(0.363818\)
Root analytic conductor: \(0.603173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9277571991\)
\(L(\frac12)\) \(\approx\) \(0.9277571991\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.33287933140796572523264980337, −9.867949724504038678540004354345, −8.828817228807071538831774871303, −7.61844874421815765798444582042, −6.96825860106344992526775285725, −5.93401766614052221110924209146, −5.07687980756451806703374478907, −4.02644344698241976310623600268, −2.63873397899260259435274010216, −1.07006860452449239445642996379, 2.25700220714382018263247780752, 3.09113650625619699444370032892, 4.35590957197330934294767898556, 5.35652577258407168235496086842, 6.56534628716596759399055173243, 7.30737169988048788687182442308, 8.227028236707561573164472820855, 9.113337325184312229164128218011, 9.628431917967447331825562623593, 11.07719868254123728156780357590

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