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Properties

Label 2-720-20.19-c2-0-2
Degree $2$
Conductor $720$
Sign $-0.799 - 0.599i$
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 − 24i·13-s + 30i·17-s + (7 + 24i)25-s − 40·29-s + 24i·37-s − 80·41-s − 49·49-s + 90i·53-s + 22·61-s + (−72 + 96i)65-s + 96i·73-s + (90 − 120i)85-s − 160·89-s − 144i·97-s + ⋯
L(s)  = 1  + (−0.800 − 0.600i)5-s − 1.84i·13-s + 1.76i·17-s + (0.280 + 0.959i)25-s − 1.37·29-s + 0.648i·37-s − 1.95·41-s − 0.999·49-s + 1.69i·53-s + 0.360·61-s + (−1.10 + 1.47i)65-s + 1.31i·73-s + (1.05 − 1.41i)85-s − 1.79·89-s − 1.48i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.799 - 0.599i$
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: \((2,\ 720,\ (\ :1),\ -0.799 - 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1510111049\)
\(L(\frac12)\) \(\approx\) \(0.1510111049\)
\(L(2)\) 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 \( 1 + 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 24iT - 169T^{2} \)
17 \( 1 - 30iT - 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 + 40T + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 24iT - 1.36e3T^{2} \)
41 \( 1 + 80T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 90iT - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 96iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 160T + 7.92e3T^{2} \)
97 \( 1 + 144iT - 9.40e3T^{2} \)
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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

−10.58567381589453681683167254130, −9.772849921071755587414925743544, −8.498488951425123768986098807252, −8.170720870217214908819389266368, −7.24725195195940277224427577656, −5.96211443788170302398041337858, −5.18186692749671004879431437784, −4.01312025747901072371038007184, −3.16541136003361737048044507542, −1.42379007206964611835854273576, 0.05370404114131641789546361144, 1.99044612965751565939732125247, 3.27713868654278199880183725610, 4.24468981042796098192167851609, 5.21353206309562500632628557885, 6.70357211977093094769584554882, 7.05109882814792000392565958298, 8.062422520151437574721816808780, 9.118993657539331068595900715533, 9.741633931670964928013090827582

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