Properties

Label 2-720-5.4-c5-0-55
Degree $2$
Conductor $720$
Sign $0.804 + 0.593i$
Analytic cond. $115.476$
Root an. cond. $10.7459$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (45 + 33.1i)5-s − 59.6i·7-s + 252·11-s + 119. i·13-s − 689. i·17-s + 220·19-s − 2.43e3i·23-s + (924. + 2.98e3i)25-s + 6.93e3·29-s − 6.75e3·31-s + (1.98e3 − 2.68e3i)35-s − 1.39e4i·37-s + 198·41-s − 417. i·43-s + 1.05e4i·47-s + ⋯
L(s)  = 1  + (0.804 + 0.593i)5-s − 0.460i·7-s + 0.627·11-s + 0.195i·13-s − 0.578i·17-s + 0.139·19-s − 0.959i·23-s + (0.295 + 0.955i)25-s + 1.53·29-s − 1.26·31-s + (0.273 − 0.370i)35-s − 1.67i·37-s + 0.0183·41-s − 0.0344i·43-s + 0.695i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.804 + 0.593i$
Analytic conductor: \(115.476\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ 0.804 + 0.593i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.669182204\)
\(L(\frac12)\) \(\approx\) \(2.669182204\)
\(L(\frac{7}{2})\) 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 \( 1 + (-45 - 33.1i)T \)
good7 \( 1 + 59.6iT - 1.68e4T^{2} \)
11 \( 1 - 252T + 1.61e5T^{2} \)
13 \( 1 - 119. iT - 3.71e5T^{2} \)
17 \( 1 + 689. iT - 1.41e6T^{2} \)
19 \( 1 - 220T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.93e3T + 2.05e7T^{2} \)
31 \( 1 + 6.75e3T + 2.86e7T^{2} \)
37 \( 1 + 1.39e4iT - 6.93e7T^{2} \)
41 \( 1 - 198T + 1.15e8T^{2} \)
43 \( 1 + 417. iT - 1.47e8T^{2} \)
47 \( 1 - 1.05e4iT - 2.29e8T^{2} \)
53 \( 1 + 5.82e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.46e4T + 7.14e8T^{2} \)
61 \( 1 + 5.69e3T + 8.44e8T^{2} \)
67 \( 1 + 4.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.33e4T + 1.80e9T^{2} \)
73 \( 1 + 7.09e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.19e4T + 3.07e9T^{2} \)
83 \( 1 - 6.18e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.99e3T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5iT - 8.58e9T^{2} \)
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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

−9.509824879243594892779965412343, −8.946115008338592699635025416803, −7.67347178334648190997557781558, −6.84166133765982907924657219422, −6.17321540506611517345470524712, −5.09067116896755134092841099086, −4.00282453510819375127940587782, −2.88818867417723993043865758414, −1.84908026162384818924121049253, −0.62026114143845586746120146031, 1.00945538302512885674216601645, 1.89354746575714583441968555692, 3.11554473428117728236681126926, 4.35770946052570680965135261146, 5.36520414027103962772436912856, 6.06745389451677500228793340953, 7.00659471411899864195927974180, 8.256038195068743085480254198922, 8.879766078992221920969633669209, 9.700666523348687745043247813654

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