↑

Properties

Label 2-2160-15.8-c1-0-0
Degree $2$
Conductor $2160$
Sign $-0.725 - 0.688i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
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  + (−1.22 − 1.87i)5-s + (−1.79 + 1.79i)7-s − 5.54i·11-s + (1.79 + 1.79i)13-s + (−1.87 − 1.87i)17-s − 3i·19-s + (−4.32 + 4.32i)23-s + (−2 + 4.58i)25-s + 4.38·29-s + 31-s + (5.54 + 1.15i)35-s + (−5 + 5i)37-s + 5.54i·41-s + (−3.20 − 3.20i)43-s + (0.646 + 0.646i)47-s + ⋯
L(s)  = 1  + (−0.547 − 0.836i)5-s + (−0.677 + 0.677i)7-s − 1.67i·11-s + (0.496 + 0.496i)13-s + (−0.453 − 0.453i)17-s − 0.688i·19-s + (−0.900 + 0.900i)23-s + (−0.400 + 0.916i)25-s + 0.814·29-s + 0.179·31-s + (0.937 + 0.195i)35-s + (−0.821 + 0.821i)37-s + 0.865i·41-s + (−0.489 − 0.489i)43-s + (0.0942 + 0.0942i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.725 - 0.688i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.725 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1360511492\)
\(L(\frac12)\) \(\approx\) \(0.1360511492\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.22 + 1.87i)T \)
good7 \( 1 + (1.79 - 1.79i)T - 7iT^{2} \)
11 \( 1 + 5.54iT - 11T^{2} \)
13 \( 1 + (-1.79 - 1.79i)T + 13iT^{2} \)
17 \( 1 + (1.87 + 1.87i)T + 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (4.32 - 4.32i)T - 23iT^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 5.54iT - 41T^{2} \)
43 \( 1 + (3.20 + 3.20i)T + 43iT^{2} \)
47 \( 1 + (-0.646 - 0.646i)T + 47iT^{2} \)
53 \( 1 + (0.0674 - 0.0674i)T - 53iT^{2} \)
59 \( 1 - 4.38T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (8.58 - 8.58i)T - 67iT^{2} \)
71 \( 1 + 5.54iT - 71T^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (8.70 - 8.70i)T - 83iT^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (3.58 - 3.58i)T - 97iT^{2} \)
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

−9.126385568933227545127928624279, −8.646320064251964009095658166114, −8.103259424333042746316694717321, −6.96636387213015338855280473932, −6.13480051239070892609154136043, −5.49287379006971822267560435436, −4.52359911980322141177769396894, −3.56601618506796362085186402454, −2.79314699437263260360286235908, −1.25921673186694225044250174370, 0.04954073809730819802361876213, 1.84724393808486153519398476509, 2.94287786506769692449053108319, 3.94620837995549763156511838034, 4.41380761662250372374431691069, 5.77726518206430241808883007839, 6.67194622448164182269561882945, 7.10106294105909444773966885211, 7.906925628806811580721540367517, 8.662309240478477237438433704803

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