Properties

Label 2-720-1.1-c3-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 8·7-s + 20·11-s + 22·13-s + 14·17-s − 76·19-s + 56·23-s + 25·25-s + 154·29-s − 160·31-s + 40·35-s − 162·37-s + 390·41-s − 388·43-s − 544·47-s − 279·49-s + 210·53-s − 100·55-s − 380·59-s − 794·61-s − 110·65-s + 148·67-s − 840·71-s + 858·73-s − 160·77-s − 144·79-s + 316·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.431·7-s + 0.548·11-s + 0.469·13-s + 0.199·17-s − 0.917·19-s + 0.507·23-s + 1/5·25-s + 0.986·29-s − 0.926·31-s + 0.193·35-s − 0.719·37-s + 1.48·41-s − 1.37·43-s − 1.68·47-s − 0.813·49-s + 0.544·53-s − 0.245·55-s − 0.838·59-s − 1.66·61-s − 0.209·65-s + 0.269·67-s − 1.40·71-s + 1.37·73-s − 0.236·77-s − 0.205·79-s + 0.417·83-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: \(1\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -1)\)

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 \( 1 + 8 T + p^{3} T^{2} \)
11 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 + 4 p T + p^{3} T^{2} \)
23 \( 1 - 56 T + p^{3} T^{2} \)
29 \( 1 - 154 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 - 390 T + p^{3} T^{2} \)
43 \( 1 + 388 T + p^{3} T^{2} \)
47 \( 1 + 544 T + p^{3} T^{2} \)
53 \( 1 - 210 T + p^{3} T^{2} \)
59 \( 1 + 380 T + p^{3} T^{2} \)
61 \( 1 + 794 T + p^{3} T^{2} \)
67 \( 1 - 148 T + p^{3} T^{2} \)
71 \( 1 + 840 T + p^{3} T^{2} \)
73 \( 1 - 858 T + p^{3} T^{2} \)
79 \( 1 + 144 T + p^{3} T^{2} \)
83 \( 1 - 316 T + p^{3} T^{2} \)
89 \( 1 + 1098 T + p^{3} T^{2} \)
97 \( 1 - 994 T + p^{3} T^{2} \)
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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

−9.515667090970250211954875376114, −8.735940617347171067390770652274, −7.920287662049133745972887999438, −6.83516597068530111860436125305, −6.19591113143144631185531053022, −4.94401713748482273728944143210, −3.92733888274160293054879181103, −3.00251305547049181151302636583, −1.47791987049911299865215679271, 0, 1.47791987049911299865215679271, 3.00251305547049181151302636583, 3.92733888274160293054879181103, 4.94401713748482273728944143210, 6.19591113143144631185531053022, 6.83516597068530111860436125305, 7.920287662049133745972887999438, 8.735940617347171067390770652274, 9.515667090970250211954875376114

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