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Properties

Label 2-2880-80.59-c0-0-0
Degree $2$
Conductor $2880$
Sign $-0.382 - 0.923i$
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  + (−0.707 + 0.707i)5-s + 1.41i·17-s + (−1 − i)19-s + 1.41i·23-s − 1.00i·25-s + 2i·31-s − 1.41·47-s + 49-s + (−1 + i)61-s + (−1.41 + 1.41i)83-s + (−1.00 − 1.00i)85-s + 1.41·95-s + (1.41 + 1.41i)107-s + (−1 + i)109-s − 1.41i·113-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + 1.41i·17-s + (−1 − i)19-s + 1.41i·23-s − 1.00i·25-s + 2i·31-s − 1.41·47-s + 49-s + (−1 + i)61-s + (−1.41 + 1.41i)83-s + (−1.00 − 1.00i)85-s + 1.41·95-s + (1.41 + 1.41i)107-s + (−1 + i)109-s − 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7599698121\)
\(L(\frac12)\) \(\approx\) \(0.7599698121\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.009205017016937245064315289019, −8.434098737880985655711942347351, −7.67002808866559835394140244406, −6.90102418570606520036168071546, −6.33991471649824408420453980283, −5.34536198699638453747668737923, −4.36033224151136259899205857334, −3.61863959614059883204302410386, −2.79637384413042231503105127177, −1.57985382659527447613009622686, 0.47495247243934691030232531325, 1.97009753626015196969253287083, 3.07560481132494690530229411026, 4.17916156963403391562130161653, 4.62628912855151761824673720445, 5.61918525179796089329292847755, 6.44092920164108307002060176671, 7.35755499035991571675236450104, 8.029439484359320539245761121457, 8.626326979959014193365358513361

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