Properties

Label 2-2880-5.4-c1-0-52
Degree $2$
Conductor $2880$
Sign $-0.894 - 0.447i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − i)5-s − 4i·13-s + 2i·17-s + (3 + 4i)25-s − 4·29-s − 12i·37-s − 8·41-s + 7·49-s + 14i·53-s − 10·61-s + (−4 + 8i)65-s + 16i·73-s + (2 − 4i)85-s − 16·89-s + 8i·97-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)5-s − 1.10i·13-s + 0.485i·17-s + (0.600 + 0.800i)25-s − 0.742·29-s − 1.97i·37-s − 1.24·41-s + 49-s + 1.92i·53-s − 1.28·61-s + (−0.496 + 0.992i)65-s + 1.87i·73-s + (0.216 − 0.433i)85-s − 1.69·89-s + 0.812i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.323587482551749396540417152965, −7.61897113993933703535736158824, −7.06160305429228138953223555934, −5.85758768358928873174161019012, −5.31575906894959062907012800200, −4.26631858909385865003102086311, −3.64228590446955423322009395276, −2.64025610554223061129127365949, −1.25244491568030934726458751790, 0, 1.61197091220539638492840631325, 2.81217713490739713776022630972, 3.65499272545099430131064274293, 4.45576074127851331169903469618, 5.22419444059934762524224117609, 6.44941257269850823609210940526, 6.88589820537729023502123313867, 7.68906851877904763743470860461, 8.398068155268665612867394609022

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