Properties

Label 2-6e4-36.31-c0-0-0
Degree $2$
Conductor $1296$
Sign $-0.642 - 0.766i$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
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  + (−0.866 + 1.5i)5-s + (−0.5 + 0.866i)13-s − 1.73·17-s + (−1 − 1.73i)25-s + (0.866 + 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (−0.866 − 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s + 1.73·89-s + (1 + 1.73i)97-s + 109-s + (0.866 − 1.5i)113-s + ⋯
L(s)  = 1  + (−0.866 + 1.5i)5-s + (−0.5 + 0.866i)13-s − 1.73·17-s + (−1 − 1.73i)25-s + (0.866 + 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (−0.866 − 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s + 1.73·89-s + (1 + 1.73i)97-s + 109-s + (0.866 − 1.5i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6381682878\)
\(L(\frac12)\) \(\approx\) \(0.6381682878\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.41793029983317460263587001971, −9.286983690778750795960351663191, −8.523071616805651160533726038411, −7.50413938217423730809569488293, −6.82678797575246410266089204738, −6.42978097528950900174302718348, −4.91118205042220952602071082669, −4.05091185198620093522574131547, −3.09721079523765228195275203066, −2.14564584167522378567864343335, 0.52240757343078394694957486898, 2.15186777005627048208888998857, 3.60036475798354218080354546801, 4.57977859764725086747004037325, 5.04473620360481570901459270148, 6.20230695646714002540597864779, 7.27080429555629325172474980546, 8.161390574520834560190547474342, 8.593591867892220266455625611607, 9.411317547906686214372234070467

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