Properties

Label 2-720-1.1-c3-0-7
Degree $2$
Conductor $720$
Sign $1$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 20·7-s + 16·11-s + 58·13-s − 38·17-s − 4·19-s − 80·23-s + 25·25-s − 82·29-s + 8·31-s − 100·35-s + 426·37-s + 246·41-s + 524·43-s − 464·47-s + 57·49-s + 702·53-s + 80·55-s − 592·59-s + 574·61-s + 290·65-s + 172·67-s + 768·71-s − 558·73-s − 320·77-s − 408·79-s + 164·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.07·7-s + 0.438·11-s + 1.23·13-s − 0.542·17-s − 0.0482·19-s − 0.725·23-s + 1/5·25-s − 0.525·29-s + 0.0463·31-s − 0.482·35-s + 1.89·37-s + 0.937·41-s + 1.85·43-s − 1.44·47-s + 0.166·49-s + 1.81·53-s + 0.196·55-s − 1.30·59-s + 1.20·61-s + 0.553·65-s + 0.313·67-s + 1.28·71-s − 0.894·73-s − 0.473·77-s − 0.581·79-s + 0.216·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.974523902\)
\(L(\frac12)\) \(\approx\) \(1.974523902\)
\(L(\frac{5}{2})\) 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 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 - 16 T + p^{3} T^{2} \)
13 \( 1 - 58 T + p^{3} T^{2} \)
17 \( 1 + 38 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 + 80 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 - 8 T + p^{3} T^{2} \)
37 \( 1 - 426 T + p^{3} T^{2} \)
41 \( 1 - 6 p T + p^{3} T^{2} \)
43 \( 1 - 524 T + p^{3} T^{2} \)
47 \( 1 + 464 T + p^{3} T^{2} \)
53 \( 1 - 702 T + p^{3} T^{2} \)
59 \( 1 + 592 T + p^{3} T^{2} \)
61 \( 1 - 574 T + p^{3} T^{2} \)
67 \( 1 - 172 T + p^{3} T^{2} \)
71 \( 1 - 768 T + p^{3} T^{2} \)
73 \( 1 + 558 T + p^{3} T^{2} \)
79 \( 1 + 408 T + p^{3} T^{2} \)
83 \( 1 - 164 T + p^{3} T^{2} \)
89 \( 1 - 510 T + p^{3} T^{2} \)
97 \( 1 - 514 T + p^{3} T^{2} \)
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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

−9.858065545698647938883200775403, −9.280220609094650233947033579541, −8.441006901583749407226096959766, −7.30117222374947947728527359042, −6.21671261941103628119253980584, −5.93681633162513379633289686726, −4.36074333727129687971051533944, −3.46827412736610572030487662709, −2.25838640691428346939414929285, −0.810234498979280370064229389079, 0.810234498979280370064229389079, 2.25838640691428346939414929285, 3.46827412736610572030487662709, 4.36074333727129687971051533944, 5.93681633162513379633289686726, 6.21671261941103628119253980584, 7.30117222374947947728527359042, 8.441006901583749407226096959766, 9.280220609094650233947033579541, 9.858065545698647938883200775403

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