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Properties

Label 2-720-5.4-c3-0-39
Degree $2$
Conductor $720$
Sign $-0.999 + 0.0160i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 + 0.178i)5-s − 33.0i·7-s + 48.3·11-s − 60.3i·13-s − 17.7i·17-s − 130.·19-s + 70.8i·23-s + (124. − 4i)25-s + 104.·29-s + 210.·31-s + (5.91 + 369. i)35-s − 300. i·37-s − 240.·41-s + 108i·43-s + 278. i·47-s + ⋯
L(s)  = 1  + (−0.999 + 0.0160i)5-s − 1.78i·7-s + 1.32·11-s − 1.28i·13-s − 0.253i·17-s − 1.58·19-s + 0.642i·23-s + (0.999 − 0.0320i)25-s + 0.669·29-s + 1.21·31-s + (0.0285 + 1.78i)35-s − 1.33i·37-s − 0.914·41-s + 0.383i·43-s + 0.865i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.0160i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -0.999 + 0.0160i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8348923813\)
\(L(\frac12)\) \(\approx\) \(0.8348923813\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 - 0.178i)T \)
good7 \( 1 + 33.0iT - 343T^{2} \)
11 \( 1 - 48.3T + 1.33e3T^{2} \)
13 \( 1 + 60.3iT - 2.19e3T^{2} \)
17 \( 1 + 17.7iT - 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 - 70.8iT - 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 - 210.T + 2.97e4T^{2} \)
37 \( 1 + 300. iT - 5.06e4T^{2} \)
41 \( 1 + 240.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 - 278. iT - 1.03e5T^{2} \)
53 \( 1 + 328. iT - 1.48e5T^{2} \)
59 \( 1 + 889.T + 2.05e5T^{2} \)
61 \( 1 + 241.T + 2.26e5T^{2} \)
67 \( 1 - 103. iT - 3.00e5T^{2} \)
71 \( 1 + 277.T + 3.57e5T^{2} \)
73 \( 1 - 274. iT - 3.89e5T^{2} \)
79 \( 1 - 366.T + 4.93e5T^{2} \)
83 \( 1 - 57.7iT - 5.71e5T^{2} \)
89 \( 1 + 203.T + 7.04e5T^{2} \)
97 \( 1 + 1.28e3iT - 9.12e5T^{2} \)
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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

−9.722963876750248014501003829146, −8.527539733583885402607388057768, −7.81046136291251474090135516362, −7.03955277545942312337408114436, −6.27551607929691410464551412361, −4.64804818044102171641110646498, −4.03501802788209472215690036116, −3.18953802459009405909252192336, −1.20715008733858410622281055657, −0.25635100082430082053659258641, 1.64984357594759028793312817325, 2.82084832260829704280764624095, 4.10965337455344709415282857066, 4.81782561469535080312531744135, 6.40742738594713612454775362567, 6.54708497883053032905411925181, 8.163470313299521446293630732524, 8.734897617098030413353748499978, 9.248976951588977247670257459722, 10.52805041045991899807444720106

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