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Properties

Label 2-720-20.19-c2-0-3
Degree $2$
Conductor $720$
Sign $-0.919 - 0.392i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 3i)5-s + 3.46·7-s − 3.46i·11-s − 6i·13-s + 18i·17-s + 13.8i·19-s − 6.92·23-s + (7 − 24i)25-s − 28·29-s + 48.4i·31-s + (−13.8 + 10.3i)35-s − 30i·37-s − 2·41-s − 62.3·43-s − 55.4·47-s + ⋯
L(s)  = 1  + (−0.800 + 0.600i)5-s + 0.494·7-s − 0.314i·11-s − 0.461i·13-s + 1.05i·17-s + 0.729i·19-s − 0.301·23-s + (0.280 − 0.959i)25-s − 0.965·29-s + 1.56i·31-s + (−0.395 + 0.296i)35-s − 0.810i·37-s − 0.0487·41-s − 1.45·43-s − 1.17·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ -0.919 - 0.392i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5646994018\)
\(L(\frac12)\) \(\approx\) \(0.5646994018\)
\(L(2)\) 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 \( 1 + (4 - 3i)T \)
good7 \( 1 - 3.46T + 49T^{2} \)
11 \( 1 + 3.46iT - 121T^{2} \)
13 \( 1 + 6iT - 169T^{2} \)
17 \( 1 - 18iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + 6.92T + 529T^{2} \)
29 \( 1 + 28T + 841T^{2} \)
31 \( 1 - 48.4iT - 961T^{2} \)
37 \( 1 + 30iT - 1.36e3T^{2} \)
41 \( 1 + 2T + 1.68e3T^{2} \)
43 \( 1 + 62.3T + 1.84e3T^{2} \)
47 \( 1 + 55.4T + 2.20e3T^{2} \)
53 \( 1 + 102iT - 2.80e3T^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 + 96.9T + 4.48e3T^{2} \)
71 \( 1 - 6.92iT - 5.04e3T^{2} \)
73 \( 1 - 132iT - 5.32e3T^{2} \)
79 \( 1 - 103. iT - 6.24e3T^{2} \)
83 \( 1 - 117.T + 6.88e3T^{2} \)
89 \( 1 - 14T + 7.92e3T^{2} \)
97 \( 1 + 24iT - 9.40e3T^{2} \)
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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

−10.65937747791654854796476766206, −9.934651375807434083794516905082, −8.549536670829549629125088396510, −8.097956461286962583103898298221, −7.17423019244942965620876426103, −6.21642031527777616489226316335, −5.17510636622496800536972186639, −3.96648652867301586078960601706, −3.18838228490656428972233363266, −1.65088139158424637742698326513, 0.19628945531262993470294254129, 1.74212986364623220156922886584, 3.24086455862927495421069637014, 4.49187887872670521108278987146, 4.98638146677466146881120208418, 6.33619822856741552008586386142, 7.44880672766954705033482671495, 7.973049587284638204204255670356, 9.056837631158755050813217789726, 9.606396380395079037075115522952

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