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Properties

Label 2-2160-240.29-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.145510677\)
\(L(\frac12)\) \(\approx\) \(1.145510677\)
\(L(1)\) 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

−9.208712827412928105776574307956, −8.357204361920694281803123970730, −7.893110472461591857894419402662, −7.04612737900362720761430417232, −5.48244397069680638827321513553, −4.85402560668873980919629450803, −4.26955174502167684606224047712, −3.40287507272241670787520641834, −1.94436799820489478805099549287, −1.23837974652002235132161079893, 1.08810651104447672285511050984, 2.94775284821295146102062262321, 3.81055102782744502031573539565, 4.80282883277623440770059015313, 5.42331498553022058175997726393, 6.34033660299625330191893121681, 7.34909997820923131555634725469, 7.74758212166572025151152236479, 8.270407491189313626747173264076, 9.242832497347617125344601510339

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