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Properties

Label 2-720-5.3-c2-0-12
Degree $2$
Conductor $720$
Sign $0.793 - 0.608i$
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  + (4.67 + 1.77i)5-s + (−3.44 + 3.44i)7-s + 11.3·11-s + (−5.55 − 5.55i)13-s + (17.3 − 17.3i)17-s + 8.69i·19-s + (11.5 + 11.5i)23-s + (18.6 + 16.5i)25-s + 35.1i·29-s − 10.6·31-s + (−22.2 + 9.99i)35-s + (−6.04 + 6.04i)37-s − 0.696·41-s + (26.4 + 26.4i)43-s + (44.2 − 44.2i)47-s + ⋯
L(s)  = 1  + (0.934 + 0.355i)5-s + (−0.492 + 0.492i)7-s + 1.03·11-s + (−0.426 − 0.426i)13-s + (1.02 − 1.02i)17-s + 0.457i·19-s + (0.502 + 0.502i)23-s + (0.747 + 0.663i)25-s + 1.21i·29-s − 0.345·31-s + (−0.635 + 0.285i)35-s + (−0.163 + 0.163i)37-s − 0.0169·41-s + (0.616 + 0.616i)43-s + (0.941 − 0.941i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.793 - 0.608i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.208572117\)
\(L(\frac12)\) \(\approx\) \(2.208572117\)
\(L(2)\) 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 \( 1 + (-4.67 - 1.77i)T \)
good7 \( 1 + (3.44 - 3.44i)T - 49iT^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 + (5.55 + 5.55i)T + 169iT^{2} \)
17 \( 1 + (-17.3 + 17.3i)T - 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (-11.5 - 11.5i)T + 529iT^{2} \)
29 \( 1 - 35.1iT - 841T^{2} \)
31 \( 1 + 10.6T + 961T^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 + 0.696T + 1.68e3T^{2} \)
43 \( 1 + (-26.4 - 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.696 - 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (-45.1 + 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 - 24.5i)T - 9.40e3iT^{2} \)
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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

−10.12183642988028949897553204921, −9.458645589985686508127887340562, −8.891847310732633595048103643445, −7.51871536503243756649392397357, −6.75164363371496396275864148867, −5.80205428290432154844952443285, −5.11853925612595649880833132676, −3.53756704873781882994952578496, −2.64257411825442332730202777292, −1.24709273617010120942840450320, 0.919700149875209498936954896984, 2.18135543878016811201394172803, 3.60779389707662713366094841090, 4.59727931034175964182317067101, 5.79207816325653523551212962519, 6.48591530529845118726979117616, 7.39089369640171370427919078246, 8.598251933303810070219684731488, 9.367176449561648347767978301712, 9.997112999725281260894279942809

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