↑

Properties

Label 2-2184-13.9-c1-0-20
Degree $2$
Conductor $2184$
Sign $0.557 - 0.829i$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
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  + (−0.5 + 0.866i)3-s + 2.52·5-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (1.56 − 2.70i)11-s + (2.38 + 2.70i)13-s + (−1.26 + 2.18i)15-s + (2.46 + 4.26i)17-s + (−0.270 − 0.467i)19-s − 0.999·21-s + (−3.22 + 5.58i)23-s + 1.36·25-s + 0.999·27-s + (2.08 − 3.61i)29-s − 1.38·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 1.12·5-s + (0.188 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.470 − 0.814i)11-s + (0.662 + 0.749i)13-s + (−0.325 + 0.564i)15-s + (0.597 + 1.03i)17-s + (−0.0619 − 0.107i)19-s − 0.218·21-s + (−0.672 + 1.16i)23-s + 0.273·25-s + 0.192·27-s + (0.387 − 0.671i)29-s − 0.248·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.557 - 0.829i$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2184} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ 0.557 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.161533285\)
\(L(\frac12)\) \(\approx\) \(2.161533285\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.38 - 2.70i)T \)
good5 \( 1 - 2.52T + 5T^{2} \)
11 \( 1 + (-1.56 + 2.70i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.46 - 4.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.270 + 0.467i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.22 - 5.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.08 + 3.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 + (-3.78 + 6.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.28 - 5.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.10 + 1.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 - 7.20T + 53T^{2} \)
59 \( 1 + (0.336 + 0.582i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.79 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.71 + 4.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.25 + 5.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.57T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.22T + 83T^{2} \)
89 \( 1 + (-2.77 + 4.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.76 - 3.06i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.153979916866220586743968753195, −8.692830288501619422875464992848, −7.71938736233978684396171134391, −6.52030165511126575800478242213, −5.83452131851882954976455283461, −5.58137551266325629879998278920, −4.23908976570680755226910213764, −3.55801835770652989469139448696, −2.24373084842826278144434619376, −1.26227389922563080006438277933, 0.891946673570879958386183040469, 1.89699469023594592197727778554, 2.86164742340685924978442298138, 4.14245645062830922619063827269, 5.15360653929288173781763057188, 5.79870639569891566351982707056, 6.61748145375693047304458046305, 7.23228943578113498509446222670, 8.169033670839741974646476309057, 8.916198730624946041899481210007

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