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Properties

Label 2-720-20.19-c0-0-0
Degree $2$
Conductor $720$
Sign $1$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s − 2·89-s − 2·101-s − 2·109-s + ⋯
L(s)  = 1  + 5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s − 2·89-s − 2·101-s − 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (559, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100269086\)
\(L(\frac12)\) \(\approx\) \(1.100269086\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.65393845174372534363834628703, −9.576965669747807839813673320093, −9.198730912360012737135715031000, −8.023579868542236758654758581506, −7.08634379863268135930907075849, −6.07173207773908522030856944187, −5.39183877784557644532980980512, −4.21033117897709809427824631381, −2.87184327636628604714433286324, −1.67160944678172248872900546019, 1.67160944678172248872900546019, 2.87184327636628604714433286324, 4.21033117897709809427824631381, 5.39183877784557644532980980512, 6.07173207773908522030856944187, 7.08634379863268135930907075849, 8.023579868542236758654758581506, 9.198730912360012737135715031000, 9.576965669747807839813673320093, 10.65393845174372534363834628703

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