↑

Properties

Label 2-2160-540.259-c0-0-0
Degree $2$
Conductor $2160$
Sign $-0.286 - 0.957i$
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  + (−0.342 + 0.939i)3-s + (0.939 + 0.342i)5-s + (−0.118 + 0.673i)7-s + (−0.766 − 0.642i)9-s + (−0.642 + 0.766i)15-s + (−0.592 − 0.342i)21-s + (0.342 + 1.93i)23-s + (0.766 + 0.642i)25-s + (0.866 − 0.500i)27-s + (0.266 + 0.223i)29-s + (−0.342 + 0.592i)35-s + (−1.17 + 0.984i)41-s + (−1.62 + 0.592i)43-s + (−0.500 − 0.866i)45-s + (0.223 − 1.26i)47-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (0.939 + 0.342i)5-s + (−0.118 + 0.673i)7-s + (−0.766 − 0.642i)9-s + (−0.642 + 0.766i)15-s + (−0.592 − 0.342i)21-s + (0.342 + 1.93i)23-s + (0.766 + 0.642i)25-s + (0.866 − 0.500i)27-s + (0.266 + 0.223i)29-s + (−0.342 + 0.592i)35-s + (−1.17 + 0.984i)41-s + (−1.62 + 0.592i)43-s + (−0.500 − 0.866i)45-s + (0.223 − 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.286 - 0.957i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133832879\)
\(L(\frac12)\) \(\approx\) \(1.133832879\)
\(L(1)\) 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 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
good7 \( 1 + (0.118 - 0.673i)T + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.342 - 1.93i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (1.62 - 0.592i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.223 + 1.26i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.984 + 0.826i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.984 + 0.826i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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

−9.662532443178729650243514321609, −8.958084720214020341165951923688, −8.154823034755812504698521122734, −6.90168000698136965360800692428, −6.24047935860089564074495566954, −5.36848516512516263421487212331, −5.01803246471335710243805220439, −3.62621923755346861876308476053, −2.94911260757529989356755241765, −1.71554507567803433547964186174, 0.850911249803931607379484232795, 1.98036372919563600821286469351, 2.89600719839146255540438677359, 4.32622138639831029589516058108, 5.19024867618828436589856423656, 5.98820897792256346586912094652, 6.74818368207095546120257776115, 7.22608433731967072047317626543, 8.446929964068908333451221090797, 8.724023316259226925106878354004

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