↑

Properties

Label 2-2160-540.439-c0-0-1
Degree $2$
Conductor $2160$
Sign $-0.973 - 0.230i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + (−0.984 + 0.173i)3-s + (−0.173 − 0.984i)5-s + (−1.50 − 1.26i)7-s + (0.939 − 0.342i)9-s + (0.342 + 0.939i)15-s + (1.70 + 0.984i)21-s + (0.984 − 0.826i)23-s + (−0.939 + 0.342i)25-s + (−0.866 + 0.5i)27-s + (−1.43 + 0.524i)29-s + (−0.984 + 1.70i)35-s + (−1.76 − 0.642i)41-s + (−0.300 + 1.70i)43-s + (−0.5 − 0.866i)45-s + (−0.524 − 0.439i)47-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (−0.173 − 0.984i)5-s + (−1.50 − 1.26i)7-s + (0.939 − 0.342i)9-s + (0.342 + 0.939i)15-s + (1.70 + 0.984i)21-s + (0.984 − 0.826i)23-s + (−0.939 + 0.342i)25-s + (−0.866 + 0.5i)27-s + (−1.43 + 0.524i)29-s + (−0.984 + 1.70i)35-s + (−1.76 − 0.642i)41-s + (−0.300 + 1.70i)43-s + (−0.5 − 0.866i)45-s + (−0.524 − 0.439i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.973 - 0.230i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.973 - 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2241229949\)
\(L(\frac12)\) \(\approx\) \(0.2241229949\)
\(L(1)\) 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 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
good7 \( 1 + (1.50 + 1.26i)T + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.984 + 0.826i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.300 - 1.70i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.524 + 0.439i)T + (0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.642 - 0.233i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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

−9.130865469510557409195132271065, −7.980045867748364197710405571329, −7.01441892389322604940574224772, −6.61974766429534458146724522725, −5.63756771119429266731417504280, −4.81692892635198273972590170165, −4.03037498334584474656407912908, −3.28576274839999456020536455521, −1.35622847263417918531691723118, −0.18349607836603787836720591651, 2.00084399851623204559701922521, 3.06603541857047628462293976333, 3.80519210449256286545408730117, 5.28291017738098782140059186882, 5.76328860477865689060936228415, 6.65887512100692139789223361952, 6.94221944254312587966429503515, 7.992639226427895059019946965457, 9.161398269905403828561820827144, 9.712408039399810942658341784501

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