Properties

Label 2-1440-1.1-c3-0-45
Degree $2$
Conductor $1440$
Sign $-1$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 16·7-s − 24·11-s − 14·13-s + 18·17-s + 36·19-s − 104·23-s + 25·25-s + 250·29-s − 28·31-s − 80·35-s − 54·37-s − 354·41-s + 228·43-s − 408·47-s − 87·49-s − 262·53-s + 120·55-s + 64·59-s + 374·61-s + 70·65-s + 300·67-s − 1.01e3·71-s + 274·73-s − 384·77-s + 788·79-s + 396·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.863·7-s − 0.657·11-s − 0.298·13-s + 0.256·17-s + 0.434·19-s − 0.942·23-s + 1/5·25-s + 1.60·29-s − 0.162·31-s − 0.386·35-s − 0.239·37-s − 1.34·41-s + 0.808·43-s − 1.26·47-s − 0.253·49-s − 0.679·53-s + 0.294·55-s + 0.141·59-s + 0.785·61-s + 0.133·65-s + 0.547·67-s − 1.69·71-s + 0.439·73-s − 0.568·77-s + 1.12·79-s + 0.523·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(84.9627\)
Root analytic conductor: \(9.21752\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
13 \( 1 + 14 T + p^{3} T^{2} \)
17 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 - 36 T + p^{3} T^{2} \)
23 \( 1 + 104 T + p^{3} T^{2} \)
29 \( 1 - 250 T + p^{3} T^{2} \)
31 \( 1 + 28 T + p^{3} T^{2} \)
37 \( 1 + 54 T + p^{3} T^{2} \)
41 \( 1 + 354 T + p^{3} T^{2} \)
43 \( 1 - 228 T + p^{3} T^{2} \)
47 \( 1 + 408 T + p^{3} T^{2} \)
53 \( 1 + 262 T + p^{3} T^{2} \)
59 \( 1 - 64 T + p^{3} T^{2} \)
61 \( 1 - 374 T + p^{3} T^{2} \)
67 \( 1 - 300 T + p^{3} T^{2} \)
71 \( 1 + 1016 T + p^{3} T^{2} \)
73 \( 1 - 274 T + p^{3} T^{2} \)
79 \( 1 - 788 T + p^{3} T^{2} \)
83 \( 1 - 396 T + p^{3} T^{2} \)
89 \( 1 + 786 T + p^{3} T^{2} \)
97 \( 1 + 1086 T + p^{3} T^{2} \)
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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

−8.467329379983277083320153500468, −8.069973264472887356272052356758, −7.29392201834200258595744675797, −6.32941905717651427389945483809, −5.22175418206935184668768722913, −4.66991757526515338218738024816, −3.56719507961430006996569248390, −2.50753651099500834346477758792, −1.33999268779207022759395489012, 0, 1.33999268779207022759395489012, 2.50753651099500834346477758792, 3.56719507961430006996569248390, 4.66991757526515338218738024816, 5.22175418206935184668768722913, 6.32941905717651427389945483809, 7.29392201834200258595744675797, 8.069973264472887356272052356758, 8.467329379983277083320153500468

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