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Properties

Label 2-2880-15.8-c1-0-39
Degree $2$
Conductor $2880$
Sign $-0.662 + 0.749i$
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  + (1.58 + 1.58i)5-s + (−1.23 + 1.23i)7-s − 1.74i·11-s + (−0.236 − 0.236i)13-s + (−4.57 − 4.57i)17-s − 6.47i·19-s + (−2.82 + 2.82i)23-s + 5.00i·25-s + 0.333·29-s − 10.4·31-s − 3.90·35-s + (2.23 − 2.23i)37-s + 7.07i·41-s + (−6.47 − 6.47i)43-s + (−4.57 − 4.57i)47-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)5-s + (−0.467 + 0.467i)7-s − 0.527i·11-s + (−0.0654 − 0.0654i)13-s + (−1.10 − 1.10i)17-s − 1.48i·19-s + (−0.589 + 0.589i)23-s + 1.00i·25-s + 0.0619·29-s − 1.88·31-s − 0.660·35-s + (0.367 − 0.367i)37-s + 1.10i·41-s + (−0.986 − 0.986i)43-s + (−0.667 − 0.667i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4876707696\)
\(L(\frac12)\) \(\approx\) \(0.4876707696\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 1.58i)T \)
good7 \( 1 + (1.23 - 1.23i)T - 7iT^{2} \)
11 \( 1 + 1.74iT - 11T^{2} \)
13 \( 1 + (0.236 + 0.236i)T + 13iT^{2} \)
17 \( 1 + (4.57 + 4.57i)T + 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 0.333T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (-2.23 + 2.23i)T - 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (6.47 + 6.47i)T + 43iT^{2} \)
47 \( 1 + (4.57 + 4.57i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 7.40T + 59T^{2} \)
61 \( 1 + 1.52T + 61T^{2} \)
67 \( 1 + (10.4 - 10.4i)T - 67iT^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (9.47 + 9.47i)T + 73iT^{2} \)
79 \( 1 - 5.52iT - 79T^{2} \)
83 \( 1 + (-7.40 + 7.40i)T - 83iT^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-1 + i)T - 97iT^{2} \)
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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

−8.830654225553122330423196469214, −7.52647374519835929359738652306, −6.94606467584749206227709837859, −6.24323186850540323763422347962, −5.51225529702985340973367155899, −4.71510875033166848863064760875, −3.44525341180933736219992507094, −2.74202428480254912791725834808, −1.91225431488371080411864340996, −0.14041165825301966064027344700, 1.50607847105589886108979679644, 2.20625928083341865975257196937, 3.65843181948353053225344189669, 4.28098720053769886054744639792, 5.22009998536319388927705322735, 6.08664366181507892195605147315, 6.59700893308583588392471018659, 7.60376316609339819029015839792, 8.390778110755434622321759211029, 9.026130797059447885420005824945

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