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Properties

Label 2-2880-120.29-c0-0-3
Degree $2$
Conductor $2880$
Sign $0.985 - 0.169i$
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  − i·5-s + 1.41i·7-s − 1.41·11-s + 1.41·13-s + 2·23-s − 25-s + 1.41·35-s + 1.41·37-s + 1.41i·41-s − 1.00·49-s + 1.41i·55-s + 1.41·59-s − 1.41i·65-s − 2.00i·77-s − 1.41i·89-s + ⋯
L(s)  = 1  − i·5-s + 1.41i·7-s − 1.41·11-s + 1.41·13-s + 2·23-s − 25-s + 1.41·35-s + 1.41·37-s + 1.41i·41-s − 1.00·49-s + 1.41i·55-s + 1.41·59-s − 1.41i·65-s − 2.00i·77-s − 1.41i·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234635700\)
\(L(\frac12)\) \(\approx\) \(1.234635700\)
\(L(1)\) 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 \( 1 + iT \)
good7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
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

−8.912383366432703353756894962660, −8.333064154273610306502354771478, −7.76590830122670286931323783510, −6.53856341888398121526241206630, −5.69946142624731506957614543564, −5.25347940829994890954191872903, −4.46720627732135103556286167878, −3.19775863235923938162513071896, −2.41060109078448395570202238038, −1.15923148354225909248848091054, 0.973951173388020184084570989698, 2.47256928897597453573065392110, 3.35062120206048695465492417240, 4.02928753092474992346755664735, 5.07628913161168378683820048224, 5.97032270251997281946222035171, 6.84879292476974116498363309364, 7.34435160862324187552357367381, 8.014805460743955320146689908108, 8.864743650528478912530898166242

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