↑

Properties

Label 2-6e4-9.5-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  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + 19-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)31-s − 37-s + (1 + 1.73i)43-s + (0.5 + 0.866i)61-s + (−0.5 + 0.866i)67-s − 73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + (0.5 + 0.866i)97-s + (−0.5 + 0.866i)103-s + 2·109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + 19-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)31-s − 37-s + (1 + 1.73i)43-s + (0.5 + 0.866i)61-s + (−0.5 + 0.866i)67-s − 73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + (0.5 + 0.866i)97-s + (−0.5 + 0.866i)103-s + 2·109-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: $\chi_{1296} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017196556\)
\(L(\frac12)\) \(\approx\) \(1.017196556\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.976460439726058876802478941422, −8.968348658327949297262650249439, −7.960091180218141000908313520896, −7.43021332229877915758758437671, −6.37565511981828672964346474963, −5.69412703741520531046856888022, −4.50009399754335994416246096190, −3.64716782261660270103531063276, −2.65918547005904147011194791027, −0.956525500722701893500926710313, 1.62883214494859894648940785575, 2.90003139280413342578176578766, 3.79245552020892662679042162058, 5.02757464026046468759624831698, 5.78394001834080959642183209820, 6.67852976714919991226496073411, 7.42429584257468520538659864693, 8.607493425990316238140279698133, 9.056943556033875290193898870540, 9.857554802837019139384160546371

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