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Properties

Label 2-1440-20.7-c1-0-12
Degree $2$
Conductor $1440$
Sign $0.793 - 0.608i$
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  + (2.22 − 0.224i)5-s + (−1.44 + 1.44i)7-s + 2.44i·11-s + (2 − 2i)13-s + (0.449 + 0.449i)17-s + 2·19-s + (0.449 + 0.449i)23-s + (4.89 − i)25-s − 0.449i·29-s + 8.89i·31-s + (−2.89 + 3.55i)35-s + (2.89 + 2.89i)37-s − 4.89·41-s + (6 + 6i)43-s + (9.34 − 9.34i)47-s + ⋯
L(s)  = 1  + (0.994 − 0.100i)5-s + (−0.547 + 0.547i)7-s + 0.738i·11-s + (0.554 − 0.554i)13-s + (0.109 + 0.109i)17-s + 0.458·19-s + (0.0937 + 0.0937i)23-s + (0.979 − 0.200i)25-s − 0.0834i·29-s + 1.59i·31-s + (−0.490 + 0.600i)35-s + (0.476 + 0.476i)37-s − 0.765·41-s + (0.914 + 0.914i)43-s + (1.36 − 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945270739\)
\(L(\frac12)\) \(\approx\) \(1.945270739\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.224i)T \)
good7 \( 1 + (1.44 - 1.44i)T - 7iT^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (-0.449 - 0.449i)T + 17iT^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-0.449 - 0.449i)T + 23iT^{2} \)
29 \( 1 + 0.449iT - 29T^{2} \)
31 \( 1 - 8.89iT - 31T^{2} \)
37 \( 1 + (-2.89 - 2.89i)T + 37iT^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + (-9.34 + 9.34i)T - 47iT^{2} \)
53 \( 1 + (2.89 - 2.89i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (6.89 - 6.89i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-7.89 + 7.89i)T - 73iT^{2} \)
79 \( 1 - 3.10T + 79T^{2} \)
83 \( 1 + (-5.55 - 5.55i)T + 83iT^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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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

−9.538332899733149268306095533487, −9.014260625929050362142769213135, −8.101922894053295689050133441282, −7.04831298592213215923383339576, −6.26434153043606672627995506095, −5.54756580737387718590019593549, −4.74836876970103008867975065686, −3.38998894656547614869955172195, −2.48783365739731674744688644026, −1.29664129240195812460883137897, 0.889145791518143419807861616668, 2.24921700038443239087836542337, 3.33479139889774377640056037894, 4.27313689992625560565751424729, 5.52433586416336050860103365784, 6.12176940237948063943271081238, 6.89275401297866063594314921842, 7.78719881351750329681256838353, 8.876666988737956904459531191906, 9.427184882276224851748554723952

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