Properties

Label 2-1440-5.4-c1-0-27
Degree $2$
Conductor $1440$
Sign $-0.894 + 0.447i$
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 − 2i)5-s − 4i·7-s − 4i·13-s + 8·19-s + 4i·23-s + (−3 + 4i)25-s − 6·29-s − 8·31-s + (−8 + 4i)35-s − 4i·37-s − 6·41-s − 4i·43-s + 4i·47-s − 9·49-s − 12i·53-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s − 1.51i·7-s − 1.10i·13-s + 1.83·19-s + 0.834i·23-s + (−0.600 + 0.800i)25-s − 1.11·29-s − 1.43·31-s + (−1.35 + 0.676i)35-s − 0.657i·37-s − 0.937·41-s − 0.609i·43-s + 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101060931\)
\(L(\frac12)\) \(\approx\) \(1.101060931\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.333477879231934759220167449011, −8.209151443680084889922028916982, −7.52342135744687534022378169885, −7.13402074126162639589090427692, −5.57248287375360996119390763337, −5.10987422315487353294961914387, −3.85664316549330218630053728810, −3.41851728508801929838304482548, −1.49057005351059371121255204327, −0.45336944881666730517742830952, 1.87322452308456859788832528505, 2.85350458962345062408189968866, 3.71599304262311115499967452216, 4.96621721210875437351677184952, 5.79360653219560157313105291554, 6.64504622952495083972104400827, 7.41144368419041654986731398540, 8.265117223089157239411512886960, 9.226058171897001814659227291784, 9.610342966639099694925149433255

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