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Properties

Label 2-2880-120.77-c1-0-23
Degree $2$
Conductor $2880$
Sign $0.792 + 0.609i$
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  + (−1.94 − 1.10i)5-s + (0.880 + 0.880i)7-s + 5.78·11-s + (−3.45 − 3.45i)13-s + (0.232 − 0.232i)17-s + 2.39·19-s + (2.68 + 2.68i)23-s + (2.56 + 4.28i)25-s + 6.38i·29-s − 5.50·31-s + (−0.741 − 2.68i)35-s + (4.65 − 4.65i)37-s + 9.14i·41-s + (−7.79 − 7.79i)43-s + (0.244 − 0.244i)47-s + ⋯
L(s)  = 1  + (−0.869 − 0.493i)5-s + (0.332 + 0.332i)7-s + 1.74·11-s + (−0.959 − 0.959i)13-s + (0.0562 − 0.0562i)17-s + 0.548·19-s + (0.560 + 0.560i)23-s + (0.513 + 0.857i)25-s + 1.18i·29-s − 0.988·31-s + (−0.125 − 0.453i)35-s + (0.764 − 0.764i)37-s + 1.42i·41-s + (−1.18 − 1.18i)43-s + (0.0356 − 0.0356i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658221087\)
\(L(\frac12)\) \(\approx\) \(1.658221087\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 + 1.10i)T \)
good7 \( 1 + (-0.880 - 0.880i)T + 7iT^{2} \)
11 \( 1 - 5.78T + 11T^{2} \)
13 \( 1 + (3.45 + 3.45i)T + 13iT^{2} \)
17 \( 1 + (-0.232 + 0.232i)T - 17iT^{2} \)
19 \( 1 - 2.39T + 19T^{2} \)
23 \( 1 + (-2.68 - 2.68i)T + 23iT^{2} \)
29 \( 1 - 6.38iT - 29T^{2} \)
31 \( 1 + 5.50T + 31T^{2} \)
37 \( 1 + (-4.65 + 4.65i)T - 37iT^{2} \)
41 \( 1 - 9.14iT - 41T^{2} \)
43 \( 1 + (7.79 + 7.79i)T + 43iT^{2} \)
47 \( 1 + (-0.244 + 0.244i)T - 47iT^{2} \)
53 \( 1 + (-6.74 + 6.74i)T - 53iT^{2} \)
59 \( 1 - 2.03iT - 59T^{2} \)
61 \( 1 + 1.70iT - 61T^{2} \)
67 \( 1 + (-9.53 + 9.53i)T - 67iT^{2} \)
71 \( 1 + 9.78iT - 71T^{2} \)
73 \( 1 + (-6.07 + 6.07i)T - 73iT^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (-1.94 + 1.94i)T - 83iT^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (-13.2 - 13.2i)T + 97iT^{2} \)
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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

−8.773035913075523620008087909977, −7.909178033504981031574947536428, −7.27938453773906753273972886150, −6.56378267350633497085013801568, −5.32797826010733860820748171042, −4.96132311427524300664421368677, −3.80913359909485471694955980733, −3.26908925199998262427571357935, −1.81594730815760105570560074717, −0.71022798679062131612590740338, 0.948922692538333328528559453001, 2.20141311874258847808046652437, 3.35922839863974682917926158424, 4.19760202789413627752646002104, 4.62983063855045853728411682228, 5.93620375227709415753539271586, 6.87300012737094759217379144315, 7.14927188774585182224998873526, 8.032544652001373325751097493047, 8.877189199474393989317934918677

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