Properties

Label 2-1440-1.1-c3-0-56
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 + 12·7-s + 20·11-s − 58·13-s + 70·17-s − 92·19-s − 112·23-s + 25·25-s − 66·29-s − 108·31-s + 60·35-s − 58·37-s − 66·41-s − 388·43-s + 408·47-s − 199·49-s − 474·53-s + 100·55-s + 540·59-s + 14·61-s − 290·65-s − 276·67-s + 96·71-s − 790·73-s + 240·77-s + 308·79-s + 1.03e3·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.647·7-s + 0.548·11-s − 1.23·13-s + 0.998·17-s − 1.11·19-s − 1.01·23-s + 1/5·25-s − 0.422·29-s − 0.625·31-s + 0.289·35-s − 0.257·37-s − 0.251·41-s − 1.37·43-s + 1.26·47-s − 0.580·49-s − 1.22·53-s + 0.245·55-s + 1.19·59-s + 0.0293·61-s − 0.553·65-s − 0.503·67-s + 0.160·71-s − 1.26·73-s + 0.355·77-s + 0.438·79-s + 1.37·83-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 \( 1 - 12 T + p^{3} T^{2} \)
11 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 - 70 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 + 112 T + p^{3} T^{2} \)
29 \( 1 + 66 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 + 58 T + p^{3} T^{2} \)
41 \( 1 + 66 T + p^{3} T^{2} \)
43 \( 1 + 388 T + p^{3} T^{2} \)
47 \( 1 - 408 T + p^{3} T^{2} \)
53 \( 1 + 474 T + p^{3} T^{2} \)
59 \( 1 - 540 T + p^{3} T^{2} \)
61 \( 1 - 14 T + p^{3} T^{2} \)
67 \( 1 + 276 T + p^{3} T^{2} \)
71 \( 1 - 96 T + p^{3} T^{2} \)
73 \( 1 + 790 T + p^{3} T^{2} \)
79 \( 1 - 308 T + p^{3} T^{2} \)
83 \( 1 - 1036 T + p^{3} T^{2} \)
89 \( 1 + 1210 T + p^{3} T^{2} \)
97 \( 1 - 1426 T + p^{3} T^{2} \)
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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

−8.750994637093222399373184354182, −7.956921574021329676449584712868, −7.19157136921729882354021784364, −6.26539708445357390629459009357, −5.39656499431805541062949332228, −4.60736987552497789370267740420, −3.60384218541512496816956076978, −2.32455792500008469701780624809, −1.51564370309523047543951183487, 0, 1.51564370309523047543951183487, 2.32455792500008469701780624809, 3.60384218541512496816956076978, 4.60736987552497789370267740420, 5.39656499431805541062949332228, 6.26539708445357390629459009357, 7.19157136921729882354021784364, 7.956921574021329676449584712868, 8.750994637093222399373184354182

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