Properties

Label 2-2880-4.3-c2-0-19
Degree $2$
Conductor $2880$
Sign $-1$
Analytic cond. $78.4743$
Root an. cond. $8.85857$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·5-s + 12.3i·7-s + 11.0i·11-s − 2.82·13-s − 6.52·17-s + 27.9i·19-s + 7.90i·23-s + 5.00·25-s + 50.7·29-s + 36.3i·31-s − 27.7i·35-s + 18.9·37-s − 5.30·41-s + 45.5i·43-s − 11.7i·47-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.77i·7-s + 1.00i·11-s − 0.216·13-s − 0.383·17-s + 1.47i·19-s + 0.343i·23-s + 0.200·25-s + 1.74·29-s + 1.17i·31-s − 0.791i·35-s + 0.511·37-s − 0.129·41-s + 1.06i·43-s − 0.249i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(78.4743\)
Root analytic conductor: \(8.85857\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.337968865\)
\(L(\frac12)\) \(\approx\) \(1.337968865\)
\(L(2)\) 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 \( 1 + 2.23T \)
good7 \( 1 - 12.3iT - 49T^{2} \)
11 \( 1 - 11.0iT - 121T^{2} \)
13 \( 1 + 2.82T + 169T^{2} \)
17 \( 1 + 6.52T + 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 - 7.90iT - 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 - 36.3iT - 961T^{2} \)
37 \( 1 - 18.9T + 1.36e3T^{2} \)
41 \( 1 + 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 + 11.7iT - 2.20e3T^{2} \)
53 \( 1 - 41.1T + 2.80e3T^{2} \)
59 \( 1 - 10.7iT - 3.48e3T^{2} \)
61 \( 1 + 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 + 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6T + 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 + 99.6iT - 6.88e3T^{2} \)
89 \( 1 + 101.T + 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−8.830853172723975910678557142697, −8.335917560580566895414080046721, −7.57058660091882332050432995037, −6.62026587064162373683977965788, −5.93309734457513798116128043172, −5.08321725534977003477079580794, −4.41608674006652361791320444835, −3.23117812675103618019252359158, −2.42541325629103031811563914766, −1.49344973670192789331363840640, 0.37526258179975177899230333491, 0.925805875576685172252377035531, 2.55274790591855405682823675014, 3.47581971954528249109909999818, 4.30867533687546701303580636894, 4.82752085943032959478442046847, 6.09298564609103753549502669358, 6.85851436603056453135442244727, 7.38085032783333951021334054698, 8.194852569258082306895102460517

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