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Properties

Label 2-2880-320.259-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: $\chi_{2880} (2179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7465319483\)
\(L(\frac12)\) \(\approx\) \(0.7465319483\)
\(L(1)\) 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

−9.147440422438398157568975629264, −8.084566935717474959144231095708, −7.65295743263660599236374337951, −7.07566036851909172442353536812, −6.22972532859452443308323760151, −5.50513061612719640315113911782, −4.55511872114406698158474160012, −3.32900187131878382397546689131, −2.52536327946602328729841334265, −1.07129857757107628814233460142, 0.791990385105727121865636747523, 1.86989411602101928570010512313, 3.07989974981620753482529666162, 3.94655724625283900960227784543, 4.65750758576267347197420217654, 5.83289923426297921050153348951, 6.67065865299536684092661371576, 7.72306700529338801900340799992, 8.182948167411135463151285150499, 8.667552820969965912989572321982

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