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Properties

Label 2-720-1.1-c1-0-9
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 − 2·7-s − 6·11-s − 4·13-s − 6·17-s + 4·19-s + 25-s − 6·29-s + 4·31-s − 2·35-s + 8·37-s − 8·43-s − 3·49-s − 6·53-s − 6·55-s − 6·59-s + 2·61-s − 4·65-s + 4·67-s + 12·71-s − 10·73-s + 12·77-s + 4·79-s − 12·83-s − 6·85-s + 12·89-s + 8·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 1.80·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.31·37-s − 1.21·43-s − 3/7·49-s − 0.824·53-s − 0.809·55-s − 0.781·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s + 1.36·77-s + 0.450·79-s − 1.31·83-s − 0.650·85-s + 1.27·89-s + 0.838·91-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−9.862862342594916931060026579517, −9.401643495521236446348069958741, −8.172980925912156702359943473709, −7.37359522787901633319248676810, −6.44739734143467503320214543326, −5.41124788289643481201476653741, −4.64413693293268396460594014295, −3.06916737303741767809697908608, −2.25797943481862032806953514483, 0, 2.25797943481862032806953514483, 3.06916737303741767809697908608, 4.64413693293268396460594014295, 5.41124788289643481201476653741, 6.44739734143467503320214543326, 7.37359522787901633319248676810, 8.172980925912156702359943473709, 9.401643495521236446348069958741, 9.862862342594916931060026579517

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