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Properties

Label 2-1440-1.1-c1-0-16
Degree $2$
Conductor $1440$
Sign $-1$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 4·11-s + 6·13-s − 2·17-s + 4·19-s + 25-s − 10·29-s − 4·31-s − 4·35-s − 10·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s − 2·53-s − 4·55-s − 12·59-s − 10·61-s + 6·65-s + 12·67-s + 10·73-s + 16·77-s − 4·79-s − 4·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.718·31-s − 0.676·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s − 0.539·55-s − 1.56·59-s − 1.28·61-s + 0.744·65-s + 1.46·67-s + 1.17·73-s + 1.82·77-s − 0.450·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.253782012201284947230265166294, −8.442396255516336718082332413226, −7.43173132830364458988873885923, −6.58286241309811525198130953754, −5.86918260574488011008112081642, −5.15078869814728840038108507425, −3.63709960988391758186186169588, −3.14392786488747653870642548167, −1.76920874114796966020786025861, 0, 1.76920874114796966020786025861, 3.14392786488747653870642548167, 3.63709960988391758186186169588, 5.15078869814728840038108507425, 5.86918260574488011008112081642, 6.58286241309811525198130953754, 7.43173132830364458988873885923, 8.442396255516336718082332413226, 9.253782012201284947230265166294

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