Properties

Label 2-2880-320.299-c0-0-0
Degree $2$
Conductor $2880$
Sign $-0.290 - 0.956i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (0.831 + 0.555i)2-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (−0.195 + 0.980i)8-s + i·10-s + (−0.707 + 0.707i)16-s + (−1.17 − 1.17i)17-s + (0.324 + 0.216i)19-s + (−0.555 + 0.831i)20-s + (0.636 + 1.53i)23-s + (−0.382 + 0.923i)25-s + 1.84·31-s + (−0.980 + 0.195i)32-s + (−0.324 − 1.63i)34-s + (0.149 + 0.360i)38-s + ⋯
L(s)  = 1  + (0.831 + 0.555i)2-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (−0.195 + 0.980i)8-s + i·10-s + (−0.707 + 0.707i)16-s + (−1.17 − 1.17i)17-s + (0.324 + 0.216i)19-s + (−0.555 + 0.831i)20-s + (0.636 + 1.53i)23-s + (−0.382 + 0.923i)25-s + 1.84·31-s + (−0.980 + 0.195i)32-s + (−0.324 − 1.63i)34-s + (0.149 + 0.360i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.290 - 0.956i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.118124786\)
\(L(\frac12)\) \(\approx\) \(2.118124786\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 - 0.555i)T \)
3 \( 1 \)
5 \( 1 + (-0.555 - 0.831i)T \)
good7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
19 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 - 1.84T + T^{2} \)
37 \( 1 + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
53 \( 1 + (-0.360 + 1.81i)T + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1 + i)T - iT^{2} \)
83 \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.237095072630449863594233652869, −8.153457093049349392369444747047, −7.48844215757612943167560958088, −6.65665347357403491500596832314, −6.34522119608586999363223638444, −5.23929295788294686049838181973, −4.75282514684819111012622186551, −3.50628727946055226782221214159, −2.90492571831272795672009596549, −1.91850468992708109921835878990, 1.05663548823061641573982198502, 2.12847758209722579129625713407, 2.94999195206591428187433287131, 4.30686065255386567748813792522, 4.57810212697514006233354442214, 5.54872029461459985005018257992, 6.32257205193512444058559883844, 6.83109438720199567041994095028, 8.212126258250487917521551820358, 8.785811199864057828241907439324

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