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Properties

Label 2-1440-120.77-c1-0-8
Degree $2$
Conductor $1440$
Sign $0.0450 - 0.998i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (−0.215 + 2.22i)5-s + (2.32 + 2.32i)7-s + 5.57·11-s + (1.79 + 1.79i)13-s + (−5.56 + 5.56i)17-s + 2.61·19-s + (−4.44 − 4.44i)23-s + (−4.90 − 0.958i)25-s − 2.12i·29-s + 4.52·31-s + (−5.66 + 4.66i)35-s + (5.02 − 5.02i)37-s − 1.73i·41-s + (−1.19 − 1.19i)43-s + (−0.849 + 0.849i)47-s + ⋯
L(s)  = 1  + (−0.0963 + 0.995i)5-s + (0.877 + 0.877i)7-s + 1.68·11-s + (0.497 + 0.497i)13-s + (−1.34 + 1.34i)17-s + 0.600·19-s + (−0.927 − 0.927i)23-s + (−0.981 − 0.191i)25-s − 0.394i·29-s + 0.812·31-s + (−0.957 + 0.788i)35-s + (0.826 − 0.826i)37-s − 0.270i·41-s + (−0.182 − 0.182i)43-s + (−0.123 + 0.123i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.0450 - 0.998i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.0450 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.914788035\)
\(L(\frac12)\) \(\approx\) \(1.914788035\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.215 - 2.22i)T \)
good7 \( 1 + (-2.32 - 2.32i)T + 7iT^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
13 \( 1 + (-1.79 - 1.79i)T + 13iT^{2} \)
17 \( 1 + (5.56 - 5.56i)T - 17iT^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + (4.44 + 4.44i)T + 23iT^{2} \)
29 \( 1 + 2.12iT - 29T^{2} \)
31 \( 1 - 4.52T + 31T^{2} \)
37 \( 1 + (-5.02 + 5.02i)T - 37iT^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 + (1.19 + 1.19i)T + 43iT^{2} \)
47 \( 1 + (0.849 - 0.849i)T - 47iT^{2} \)
53 \( 1 + (4.22 - 4.22i)T - 53iT^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 - 3.13iT - 61T^{2} \)
67 \( 1 + (-1.86 + 1.86i)T - 67iT^{2} \)
71 \( 1 - 6.95iT - 71T^{2} \)
73 \( 1 + (5.86 - 5.86i)T - 73iT^{2} \)
79 \( 1 + 2.52iT - 79T^{2} \)
83 \( 1 + (-0.694 + 0.694i)T - 83iT^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-4.09 - 4.09i)T + 97iT^{2} \)
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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

−9.674262946948898347861401957609, −8.770321599481731210856598478867, −8.330334786063456668846835399191, −7.20037702210954690877111790068, −6.29396221369426254967053281386, −5.98233563361629350642587423390, −4.39632741604245209611747429404, −3.90034989090359303277807452581, −2.48536455810687830817378652029, −1.62783913139476686946615507829, 0.844980212121450216445762351068, 1.70389885265179269173244404955, 3.44576624930402248552647977406, 4.39950771348534735955401002882, 4.87270430312363090668323973362, 6.06501766140331939217680375587, 6.95883732820628001450864681731, 7.83084755060046122380093227488, 8.519345658459552751921265564038, 9.347149600775755598139800580484

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