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Properties

Label 2-2160-20.3-c1-0-16
Degree $2$
Conductor $2160$
Sign $0.957 - 0.287i$
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 + (−1.73 − 1.73i)7-s + 5.97i·11-s + (4.67 + 4.67i)13-s + (2.67 − 2.67i)17-s − 0.778·19-s + (1.34 − 1.34i)23-s + (−3 − 4i)25-s + 5i·29-s + 5.97i·31-s + (−5.19 + 1.73i)35-s + (−4 + 4i)37-s + 5.34·41-s + (−1.34 + 1.34i)43-s + (−1.34 − 1.34i)47-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + (−0.654 − 0.654i)7-s + 1.80i·11-s + (1.29 + 1.29i)13-s + (0.648 − 0.648i)17-s − 0.178·19-s + (0.279 − 0.279i)23-s + (−0.600 − 0.800i)25-s + 0.928i·29-s + 1.07i·31-s + (−0.878 + 0.292i)35-s + (−0.657 + 0.657i)37-s + 0.835·41-s + (−0.204 + 0.204i)43-s + (−0.195 − 0.195i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.843847647\)
\(L(\frac12)\) \(\approx\) \(1.843847647\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 1.73i)T + 7iT^{2} \)
11 \( 1 - 5.97iT - 11T^{2} \)
13 \( 1 + (-4.67 - 4.67i)T + 13iT^{2} \)
17 \( 1 + (-2.67 + 2.67i)T - 17iT^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
23 \( 1 + (-1.34 + 1.34i)T - 23iT^{2} \)
29 \( 1 - 5iT - 29T^{2} \)
31 \( 1 - 5.97iT - 31T^{2} \)
37 \( 1 + (4 - 4i)T - 37iT^{2} \)
41 \( 1 - 5.34T + 41T^{2} \)
43 \( 1 + (1.34 - 1.34i)T - 43iT^{2} \)
47 \( 1 + (1.34 + 1.34i)T + 47iT^{2} \)
53 \( 1 + (-5.34 - 5.34i)T + 53iT^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 1.34T + 61T^{2} \)
67 \( 1 + (4.24 + 4.24i)T + 67iT^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 + (-2 - 2i)T + 73iT^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 + (-5.97 + 5.97i)T - 83iT^{2} \)
89 \( 1 + 15.3iT - 89T^{2} \)
97 \( 1 + (-9.34 + 9.34i)T - 97iT^{2} \)
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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

−9.001761628300935870980440817236, −8.655593839887134969752865834547, −7.30695704484937267205544096489, −6.90857843269645126877640308551, −6.02273208768877757902082771863, −4.92852680105360820007762337174, −4.37897571208749787118895711250, −3.43387064194325186571451553012, −2.00191929378899596104207006722, −1.12991791689020481762361511492, 0.76439427337284980751177141631, 2.35549922276663347969194891247, 3.33406617968959347949109726284, 3.64984183282608222132489058167, 5.54426231502823792838066922742, 5.93481457025590012570471133913, 6.32753633131866775310706814050, 7.59652942230526970421381824610, 8.317866144556651477968571032684, 8.964221103693768846355717029991

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