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Properties

Label 2-1440-120.53-c1-0-18
Degree $2$
Conductor $1440$
Sign $-0.719 + 0.694i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.719 + 0.694i$
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: \((2,\ 1440,\ (\ :1/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6973799475\)
\(L(\frac12)\) \(\approx\) \(0.6973799475\)
\(L(\frac{3}{2})\) 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

−9.146495682643768968152729198670, −8.619662526242623746383274897471, −7.60811229385579072145247610004, −6.86703237221470317708526227448, −5.82775755767249824276844333709, −5.08221692714252399961306833079, −4.00751661666643795004374739723, −3.25205184238766552769877034624, −1.81308911686866886614677806480, −0.28398514088884425062436431451, 1.50773913141447083260579336350, 3.09497939457869831633233499866, 3.71300596694930911065863447714, 4.59091932389245753640627920442, 5.92316768368580475529624304886, 6.81184181215795815102841357842, 7.14838117359392354334688388659, 8.234474962259869209534618922357, 9.005123066881745497999481854616, 9.955917517075607447687197748856

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