Properties

Label 2-720-5.4-c3-0-40
Degree $2$
Conductor $720$
Sign $-0.724 + 0.688i$
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  + (8.10 − 7.70i)5-s − 22.2i·7-s − 1.79·11-s − 58.2i·13-s − 18.9i·17-s + 104.·19-s + 49.6i·23-s + (6.37 − 124. i)25-s − 293.·29-s − 64.4·31-s + (−171. − 180i)35-s − 19.8i·37-s + 165.·41-s − 247. i·43-s + 384. i·47-s + ⋯
L(s)  = 1  + (0.724 − 0.688i)5-s − 1.19i·7-s − 0.0490·11-s − 1.24i·13-s − 0.270i·17-s + 1.26·19-s + 0.449i·23-s + (0.0509 − 0.998i)25-s − 1.87·29-s − 0.373·31-s + (−0.826 − 0.869i)35-s − 0.0883i·37-s + 0.630·41-s − 0.877i·43-s + 1.19i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.688i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.724 + 0.688i$
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: \((2,\ 720,\ (\ :3/2),\ -0.724 + 0.688i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.848439930\)
\(L(\frac12)\) \(\approx\) \(1.848439930\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.10 + 7.70i)T \)
good7 \( 1 + 22.2iT - 343T^{2} \)
11 \( 1 + 1.79T + 1.33e3T^{2} \)
13 \( 1 + 58.2iT - 2.19e3T^{2} \)
17 \( 1 + 18.9iT - 4.91e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 - 49.6iT - 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 + 64.4T + 2.97e4T^{2} \)
37 \( 1 + 19.8iT - 5.06e4T^{2} \)
41 \( 1 - 165.T + 6.89e4T^{2} \)
43 \( 1 + 247. iT - 7.95e4T^{2} \)
47 \( 1 - 384. iT - 1.03e5T^{2} \)
53 \( 1 - 463. iT - 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 173. iT - 3.00e5T^{2} \)
71 \( 1 + 594.T + 3.57e5T^{2} \)
73 \( 1 + 320. iT - 3.89e5T^{2} \)
79 \( 1 + 770.T + 4.93e5T^{2} \)
83 \( 1 + 173. iT - 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 384. iT - 9.12e5T^{2} \)
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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

−9.695684310981481212772190774882, −9.011724749245625832574039782162, −7.73609341922517939699695810462, −7.32760523999665518595100334166, −5.89330648341779241396589488719, −5.28200316590202535669379322091, −4.15306207506281699291302613090, −3.04933660917032998174598126845, −1.50028072794779841463571661946, −0.50083597825191823173676834139, 1.69981276230065148776252260041, 2.54704828842364215306893214284, 3.71213817882796943753970028791, 5.16854727168092994149537940102, 5.87709887187600333241532550495, 6.74175498064494535197159883119, 7.64580665282854810885259455439, 8.936175134320503544711572672320, 9.363780269095593972204098598336, 10.22685917584071793725426803812

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