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Properties

Label 2-2880-16.13-c1-0-39
Degree $2$
Conductor $2880$
Sign $-0.923 - 0.382i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
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 − 4.82i·7-s + (1.41 − 1.41i)11-s + (−0.585 − 0.585i)13-s − 5.41·17-s + (−3.82 − 3.82i)19-s + 5.41i·23-s − 1.00i·25-s + (0.585 + 0.585i)29-s − 3.65·31-s + (3.41 + 3.41i)35-s + (4.58 − 4.58i)37-s + 4.82i·41-s + (−3.65 + 3.65i)43-s − 7.07·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s − 1.82i·7-s + (0.426 − 0.426i)11-s + (−0.162 − 0.162i)13-s − 1.31·17-s + (−0.878 − 0.878i)19-s + 1.12i·23-s − 0.200i·25-s + (0.108 + 0.108i)29-s − 0.656·31-s + (0.577 + 0.577i)35-s + (0.753 − 0.753i)37-s + 0.754i·41-s + (−0.557 + 0.557i)43-s − 1.03·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2411612824\)
\(L(\frac12)\) \(\approx\) \(0.2411612824\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.82iT - 7T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + (3.82 + 3.82i)T + 19iT^{2} \)
23 \( 1 - 5.41iT - 23T^{2} \)
29 \( 1 + (-0.585 - 0.585i)T + 29iT^{2} \)
31 \( 1 + 3.65T + 31T^{2} \)
37 \( 1 + (-4.58 + 4.58i)T - 37iT^{2} \)
41 \( 1 - 4.82iT - 41T^{2} \)
43 \( 1 + (3.65 - 3.65i)T - 43iT^{2} \)
47 \( 1 + 7.07T + 47T^{2} \)
53 \( 1 + (-4 + 4i)T - 53iT^{2} \)
59 \( 1 + (7.41 - 7.41i)T - 59iT^{2} \)
61 \( 1 + (-9.48 - 9.48i)T + 61iT^{2} \)
67 \( 1 + (-7.65 - 7.65i)T + 67iT^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 3.17iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + (3.07 + 3.07i)T + 83iT^{2} \)
89 \( 1 - 3.65iT - 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−8.255272445699174556812320987213, −7.47885034294886632250932888592, −6.87638957963966682733597490316, −6.35916235041932959127136366652, −5.08799199223083203900548798445, −4.17484155483564348684506678060, −3.77387298925688915157723342194, −2.62920416417805850310614101208, −1.26912388029076335043821907003, −0.07657363644073475144080116209, 1.88475167496763988928312597648, 2.42924184599820700324495581500, 3.68255788747597749968885381143, 4.63084691622531435835797282866, 5.25881038360730769682031326321, 6.33028208385152991339262919303, 6.62596797254418768842114041720, 7.977705340069364437796660401183, 8.492254692077510987816339139981, 9.092476159867671185292337413089

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