Properties

Label 2-1440-1.1-c3-0-52
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 + 6·7-s − 60·11-s + 50·13-s + 30·17-s + 40·19-s − 178·23-s + 25·25-s − 166·29-s + 20·31-s + 30·35-s + 10·37-s + 250·41-s + 142·43-s − 214·47-s − 307·49-s − 490·53-s − 300·55-s + 800·59-s + 250·61-s + 250·65-s − 774·67-s − 100·71-s − 230·73-s − 360·77-s − 1.32e3·79-s − 982·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.323·7-s − 1.64·11-s + 1.06·13-s + 0.428·17-s + 0.482·19-s − 1.61·23-s + 1/5·25-s − 1.06·29-s + 0.115·31-s + 0.144·35-s + 0.0444·37-s + 0.952·41-s + 0.503·43-s − 0.664·47-s − 0.895·49-s − 1.26·53-s − 0.735·55-s + 1.76·59-s + 0.524·61-s + 0.477·65-s − 1.41·67-s − 0.167·71-s − 0.368·73-s − 0.532·77-s − 1.87·79-s − 1.29·83-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 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 + 60 T + p^{3} T^{2} \)
13 \( 1 - 50 T + p^{3} T^{2} \)
17 \( 1 - 30 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 + 178 T + p^{3} T^{2} \)
29 \( 1 + 166 T + p^{3} T^{2} \)
31 \( 1 - 20 T + p^{3} T^{2} \)
37 \( 1 - 10 T + p^{3} T^{2} \)
41 \( 1 - 250 T + p^{3} T^{2} \)
43 \( 1 - 142 T + p^{3} T^{2} \)
47 \( 1 + 214 T + p^{3} T^{2} \)
53 \( 1 + 490 T + p^{3} T^{2} \)
59 \( 1 - 800 T + p^{3} T^{2} \)
61 \( 1 - 250 T + p^{3} T^{2} \)
67 \( 1 + 774 T + p^{3} T^{2} \)
71 \( 1 + 100 T + p^{3} T^{2} \)
73 \( 1 + 230 T + p^{3} T^{2} \)
79 \( 1 + 1320 T + p^{3} T^{2} \)
83 \( 1 + 982 T + p^{3} T^{2} \)
89 \( 1 + 874 T + p^{3} T^{2} \)
97 \( 1 + 310 T + p^{3} T^{2} \)
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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

−8.631110007440500643643641929041, −7.975291981124062710162686327239, −7.32040008714933576267079431405, −5.99884457062361662239430896150, −5.62401479097580832018562461052, −4.59655277249171470934803855051, −3.49836051256242586291057072221, −2.46129211346905869168313225818, −1.43201257363883792789469627823, 0, 1.43201257363883792789469627823, 2.46129211346905869168313225818, 3.49836051256242586291057072221, 4.59655277249171470934803855051, 5.62401479097580832018562461052, 5.99884457062361662239430896150, 7.32040008714933576267079431405, 7.975291981124062710162686327239, 8.631110007440500643643641929041

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