Properties

Label 2-720-1.1-c1-0-8
Degree $2$
Conductor $720$
Sign $-1$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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 − 6·13-s + 2·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s − 4·35-s − 6·37-s − 10·41-s + 4·43-s + 8·47-s + 9·49-s − 10·53-s + 6·61-s − 6·65-s + 4·67-s − 14·73-s − 16·79-s + 12·83-s + 2·85-s − 2·89-s + 24·91-s − 4·95-s + 2·97-s + 14·101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.676·35-s − 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.768·61-s − 0.744·65-s + 0.488·67-s − 1.63·73-s − 1.80·79-s + 1.31·83-s + 0.216·85-s − 0.211·89-s + 2.51·91-s − 0.410·95-s + 0.203·97-s + 1.39·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.06614419685989265635505817070, −9.345822684053937464266140209197, −8.312681871958482675893970014801, −7.18880346144848821197627995561, −6.46041242786302870623045801913, −5.61049576810701540702833296151, −4.43140184316888462520939708171, −3.20808194979498891977471116338, −2.19282779410354418054159546466, 0, 2.19282779410354418054159546466, 3.20808194979498891977471116338, 4.43140184316888462520939708171, 5.61049576810701540702833296151, 6.46041242786302870623045801913, 7.18880346144848821197627995561, 8.312681871958482675893970014801, 9.345822684053937464266140209197, 10.06614419685989265635505817070

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