Properties

Label 2-6e3-72.59-c1-0-2
Degree $2$
Conductor $216$
Sign $-0.518 - 0.855i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.867 + 1.11i)2-s + (−0.494 + 1.93i)4-s + (−0.895 + 1.55i)5-s + (−2.08 + 1.20i)7-s + (−2.59 + 1.12i)8-s + (−2.50 + 0.345i)10-s + (1.36 − 0.790i)11-s + (5.35 + 3.09i)13-s + (−3.15 − 1.28i)14-s + (−3.51 − 1.91i)16-s − 3.69i·17-s + 3.12·19-s + (−2.56 − 2.50i)20-s + (2.07 + 0.843i)22-s + (1.36 − 2.35i)23-s + ⋯
L(s)  = 1  + (0.613 + 0.789i)2-s + (−0.247 + 0.968i)4-s + (−0.400 + 0.693i)5-s + (−0.789 + 0.455i)7-s + (−0.916 + 0.398i)8-s + (−0.793 + 0.109i)10-s + (0.412 − 0.238i)11-s + (1.48 + 0.857i)13-s + (−0.843 − 0.343i)14-s + (−0.877 − 0.479i)16-s − 0.897i·17-s + 0.717·19-s + (−0.572 − 0.559i)20-s + (0.441 + 0.179i)22-s + (0.283 − 0.491i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.518 - 0.855i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.518 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689907 + 1.22467i\)
\(L(\frac12)\) \(\approx\) \(0.689907 + 1.22467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.867 - 1.11i)T \)
3 \( 1 \)
good5 \( 1 + (0.895 - 1.55i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.08 - 1.20i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 0.790i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.35 - 3.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.69iT - 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + (-1.36 + 2.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.55 - 4.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.24iT - 37T^{2} \)
41 \( 1 + (-5.32 - 3.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.452 + 0.783i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.88 + 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.05T + 53T^{2} \)
59 \( 1 + (-6.10 - 3.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.05 - 1.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 1.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.31T + 71T^{2} \)
73 \( 1 - 0.631T + 73T^{2} \)
79 \( 1 + (-7.82 + 4.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.5 - 7.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.16iT - 89T^{2} \)
97 \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.82132307482297650032448840744, −11.73512701661155897259345418429, −11.05724299216048675203914631520, −9.374121689461764676377875019547, −8.662329299693646355275058911199, −7.24882539024761439891928949617, −6.57461237892193843644102786124, −5.55290102403316960722643452950, −3.95584200430514196484730046926, −3.02683312250912332990950407405, 1.12096017564893920421714367734, 3.30459226541255685651313188307, 4.15716507546979393411359897667, 5.54982941692291816762402581676, 6.58106865057804046758441638364, 8.196036618814802314580793107936, 9.241124746027704811051257057490, 10.24549280390966512755167168053, 11.10650641952619791792098550535, 12.13916050048435805778210619898

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