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Properties

Label 2-1440-120.53-c1-0-1
Degree $2$
Conductor $1440$
Sign $-0.992 - 0.119i$
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  + (1.45 + 1.69i)5-s + (−1.53 + 1.53i)7-s − 2.72·11-s + (−0.857 + 0.857i)13-s + (−2.55 − 2.55i)17-s − 3.54·19-s + (−0.626 + 0.626i)23-s + (−0.772 + 4.93i)25-s + 5.12i·29-s − 7.89·31-s + (−4.82 − 0.375i)35-s + (4.21 + 4.21i)37-s − 12.4i·41-s + (5.67 − 5.67i)43-s + (−9.45 − 9.45i)47-s + ⋯
L(s)  = 1  + (0.650 + 0.759i)5-s + (−0.578 + 0.578i)7-s − 0.821·11-s + (−0.237 + 0.237i)13-s + (−0.619 − 0.619i)17-s − 0.812·19-s + (−0.130 + 0.130i)23-s + (−0.154 + 0.987i)25-s + 0.951i·29-s − 1.41·31-s + (−0.815 − 0.0634i)35-s + (0.693 + 0.693i)37-s − 1.93i·41-s + (0.865 − 0.865i)43-s + (−1.37 − 1.37i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.992 - 0.119i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.992 - 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5410492433\)
\(L(\frac12)\) \(\approx\) \(0.5410492433\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 1.69i)T \)
good7 \( 1 + (1.53 - 1.53i)T - 7iT^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + (0.857 - 0.857i)T - 13iT^{2} \)
17 \( 1 + (2.55 + 2.55i)T + 17iT^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 + (0.626 - 0.626i)T - 23iT^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 + (-4.21 - 4.21i)T + 37iT^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (-5.67 + 5.67i)T - 43iT^{2} \)
47 \( 1 + (9.45 + 9.45i)T + 47iT^{2} \)
53 \( 1 + (6.46 + 6.46i)T + 53iT^{2} \)
59 \( 1 - 2.51iT - 59T^{2} \)
61 \( 1 - 9.49iT - 61T^{2} \)
67 \( 1 + (-9.91 - 9.91i)T + 67iT^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + (5.71 + 5.71i)T + 73iT^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 + (-3.58 - 3.58i)T + 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (1.29 - 1.29i)T - 97iT^{2} \)
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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

−9.888155375108032437601910123971, −9.196992778829962828861203833841, −8.459764951960317856737986302169, −7.22818867807654975977165297668, −6.77428479788792139025213420139, −5.76282047752675785948601978577, −5.15635637855376998147563767351, −3.79859859411776350633577052338, −2.72983603179007658085089322477, −2.04952737968242506916560109318, 0.19555489622261923537537165890, 1.75799623296199421491822540479, 2.86155050513658244922014639282, 4.16030419082093766504045774169, 4.86587353384257371087698639216, 5.97013548793854946129054126508, 6.47099854226422386492486580026, 7.72985816764950868237707125256, 8.247604433625181934818801974592, 9.376718967232906056871879274150

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