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Properties

Label 2-2160-5.4-c1-0-16
Degree $2$
Conductor $2160$
Sign $0.894 - 0.447i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 2i)5-s + 4i·7-s + 5·11-s − 3i·13-s + i·17-s − 6·19-s + i·23-s + (−3 + 4i)25-s + 9·29-s + 5·31-s + (8 − 4i)35-s + 2i·37-s − 2·41-s − i·43-s + 13i·47-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + 1.51i·7-s + 1.50·11-s − 0.832i·13-s + 0.242i·17-s − 1.37·19-s + 0.208i·23-s + (−0.600 + 0.800i)25-s + 1.67·29-s + 0.898·31-s + (1.35 − 0.676i)35-s + 0.328i·37-s − 0.312·41-s − 0.152i·43-s + 1.89i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663058967\)
\(L(\frac12)\) \(\approx\) \(1.663058967\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 13iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.976900416331365502461194250485, −8.466009955994635674350779298181, −7.912474994140581120583008461352, −6.49753885714853443022114435659, −6.10704466593891491551517401134, −5.05789318226404338438379554810, −4.38595006375639215852257042785, −3.34711338459262275031703629110, −2.22551931996650733119819391832, −1.02952460791105264657679749735, 0.73992548487038741356440557818, 2.08790520080880437263343051638, 3.42089892076631008528000220231, 4.10565520611227391321923074628, 4.61463244384163357566221075484, 6.39814711543071832995400608678, 6.63261542824311388824443934446, 7.24576276594186805667364826961, 8.232303287040387939575464283783, 8.935287782589884645318706684027

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