Properties

Label 2-2880-15.8-c1-0-17
Degree $2$
Conductor $2880$
Sign $0.461 - 0.887i$
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  + (1.91 + 1.15i)5-s + (1.70 − 1.70i)7-s − 0.892i·11-s + (2.70 + 2.70i)13-s + (1.65 + 1.65i)17-s + 7.41i·19-s + (−6.13 + 6.13i)23-s + (2.34 + 4.41i)25-s − 0.521·29-s − 5.26·31-s + (5.24 − 1.30i)35-s + (−1.78 + 1.78i)37-s − 0.110i·41-s + (1.26 + 1.26i)43-s + (−6.77 − 6.77i)47-s + ⋯
L(s)  = 1  + (0.856 + 0.515i)5-s + (0.646 − 0.646i)7-s − 0.269i·11-s + (0.751 + 0.751i)13-s + (0.401 + 0.401i)17-s + 1.70i·19-s + (−1.27 + 1.27i)23-s + (0.468 + 0.883i)25-s − 0.0969·29-s − 0.945·31-s + (0.886 − 0.220i)35-s + (−0.293 + 0.293i)37-s − 0.0173i·41-s + (0.192 + 0.192i)43-s + (−0.987 − 0.987i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.461 - 0.887i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.461 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.273708418\)
\(L(\frac12)\) \(\approx\) \(2.273708418\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 - 1.15i)T \)
good7 \( 1 + (-1.70 + 1.70i)T - 7iT^{2} \)
11 \( 1 + 0.892iT - 11T^{2} \)
13 \( 1 + (-2.70 - 2.70i)T + 13iT^{2} \)
17 \( 1 + (-1.65 - 1.65i)T + 17iT^{2} \)
19 \( 1 - 7.41iT - 19T^{2} \)
23 \( 1 + (6.13 - 6.13i)T - 23iT^{2} \)
29 \( 1 + 0.521T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + (1.78 - 1.78i)T - 37iT^{2} \)
41 \( 1 + 0.110iT - 41T^{2} \)
43 \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \)
47 \( 1 + (6.77 + 6.77i)T + 47iT^{2} \)
53 \( 1 + (4.07 - 4.07i)T - 53iT^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-5.26 + 5.26i)T - 67iT^{2} \)
71 \( 1 + 3.05iT - 71T^{2} \)
73 \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 + (-6.54 + 6.54i)T - 83iT^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (-8.60 + 8.60i)T - 97iT^{2} \)
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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

−8.885720836020290317436938096906, −8.065041279351647580705779660808, −7.47777499069223703565123140650, −6.51425833190883953736830270680, −5.88673627411265323625367489290, −5.22356115685220082005957661916, −3.90376433181229900506841451449, −3.53051833546091528204000351671, −1.94753564935328089294227865358, −1.45392632527134597631906112261, 0.72952261331429580848836731549, 1.98562241982969328579993240174, 2.67242820170299772615839928850, 3.96858920237994710493220875123, 5.02203092690400449918897943065, 5.39489146416572638504877881641, 6.27973851239384382347390893273, 7.04478800187131322369483335032, 8.242942721325573267854659851087, 8.492465475936921768259227229403

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