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Properties

Label 2-2160-180.119-c1-0-0
Degree $2$
Conductor $2160$
Sign $-0.963 + 0.269i$
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  + (−1.74 − 1.40i)5-s + (−2.45 + 4.25i)7-s + (−1.62 + 2.81i)11-s + (3.94 − 2.27i)13-s + 4.68·17-s − 2.29i·19-s + (2.41 − 1.39i)23-s + (1.07 + 4.88i)25-s + (−0.841 − 0.485i)29-s + (−8.33 + 4.81i)31-s + (10.2 − 3.97i)35-s − 2.32i·37-s + (−8.23 + 4.75i)41-s + (0.256 − 0.443i)43-s + (−7.37 − 4.25i)47-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)5-s + (−0.928 + 1.60i)7-s + (−0.489 + 0.847i)11-s + (1.09 − 0.631i)13-s + 1.13·17-s − 0.527i·19-s + (0.504 − 0.291i)23-s + (0.215 + 0.976i)25-s + (−0.156 − 0.0902i)29-s + (−1.49 + 0.864i)31-s + (1.73 − 0.672i)35-s − 0.381i·37-s + (−1.28 + 0.742i)41-s + (0.0390 − 0.0676i)43-s + (−1.07 − 0.620i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.963 + 0.269i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.963 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06445431320\)
\(L(\frac12)\) \(\approx\) \(0.06445431320\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 + 1.40i)T \)
good7 \( 1 + (2.45 - 4.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.62 - 2.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.94 + 2.27i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + (-2.41 + 1.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.841 + 0.485i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.33 - 4.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.32iT - 37T^{2} \)
41 \( 1 + (8.23 - 4.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.256 + 0.443i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.37 + 4.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.90T + 53T^{2} \)
59 \( 1 + (-1.48 - 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.67 + 2.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.05 + 7.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.66T + 71T^{2} \)
73 \( 1 + 3.66iT - 73T^{2} \)
79 \( 1 + (-0.00584 - 0.00337i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.41 + 1.39i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.952iT - 89T^{2} \)
97 \( 1 + (3.07 + 1.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.325612230000103683521615493571, −8.729519033696529988455869280783, −8.111187327074139727634053096901, −7.23294630809490463229886536060, −6.28439389607034145753029780195, −5.41683540676888842742905406982, −4.92560744350810924313205535858, −3.52229689276064095110662779261, −3.01183978038839415132927361699, −1.63903029477795851809410553260, 0.02408794580428320557272962235, 1.29644221729536771836944315177, 3.21781205199888895311523927894, 3.51502675096889436751779774325, 4.27129906991486255692750724043, 5.63225032795171964182132720581, 6.45079265985285648678178284941, 7.14733932411314032953575007860, 7.75782563556789815788586168676, 8.489889321195856458738137683529

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