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Properties

Label 2-2160-240.29-c0-0-1
Degree $2$
Conductor $2160$
Sign $-0.382 - 0.923i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (−0.707 − 0.707i)5-s − 1.41·7-s − i·8-s + (0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 1.41i·14-s + 16-s − 17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + (−0.707 + 0.707i)22-s + i·23-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.707 − 0.707i)5-s − 1.41·7-s − i·8-s + (0.707 − 0.707i)10-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 1.41i·14-s + 16-s − 17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + (−0.707 + 0.707i)22-s + i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6896724997\)
\(L(\frac12)\) \(\approx\) \(0.6896724997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−9.316276632414323266340059930739, −8.771029177510898244414791399810, −7.76815261074276566278233011130, −7.28266607397041973032234032549, −6.33625715295151641563856967775, −5.76346771413092523889889977936, −4.74793045520512264184654670387, −3.85749179312171973910118910423, −3.28956062057192296939018032324, −1.15776006786732501871274067496, 0.58725968488981043727340413373, 2.37350128787113147952784948725, 3.20377485212611317776624811218, 3.83492817778156374844949669927, 4.59898512648147778312873068240, 6.07356121775251998160959923331, 6.55090295747870129447713192329, 7.46029537813377768107182046843, 8.657339082852255909772946286959, 9.049316368975861635188095938649

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