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Properties

Label 2-720-9.4-c1-0-8
Degree $2$
Conductor $720$
Sign $0.711 - 0.702i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 0.0696i)3-s + (−0.5 + 0.866i)5-s + (−0.165 − 0.287i)7-s + (2.99 − 0.240i)9-s + (2.20 + 3.81i)11-s + (−3.49 + 6.04i)13-s + (−0.805 + 1.53i)15-s − 3.07·17-s + 7.55·19-s + (−0.307 − 0.485i)21-s + (0.834 − 1.44i)23-s + (−0.499 − 0.866i)25-s + (5.15 − 0.625i)27-s + (−1.78 − 3.09i)29-s + (2.82 − 4.88i)31-s + ⋯
L(s)  = 1  + (0.999 − 0.0401i)3-s + (−0.223 + 0.387i)5-s + (−0.0627 − 0.108i)7-s + (0.996 − 0.0803i)9-s + (0.664 + 1.15i)11-s + (−0.968 + 1.67i)13-s + (−0.207 + 0.395i)15-s − 0.744·17-s + 1.73·19-s + (−0.0670 − 0.106i)21-s + (0.173 − 0.301i)23-s + (−0.0999 − 0.173i)25-s + (0.992 − 0.120i)27-s + (−0.331 − 0.574i)29-s + (0.506 − 0.877i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91378 + 0.785022i\)
\(L(\frac12)\) \(\approx\) \(1.91378 + 0.785022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0696i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.165 + 0.287i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.20 - 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.49 - 6.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 - 7.55T + 19T^{2} \)
23 \( 1 + (-0.834 + 1.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.78 + 3.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.82 + 4.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + (2.74 - 4.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.53 + 6.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.98 - 6.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.47T + 53T^{2} \)
59 \( 1 + (3.69 - 6.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.296 + 0.513i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.61 + 4.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.98T + 71T^{2} \)
73 \( 1 + 8.05T + 73T^{2} \)
79 \( 1 + (5.49 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.14 + 5.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-0.622 - 1.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19672321893704109935198445253, −9.484658082408332252647191862051, −9.087316373646355427482343545727, −7.69958889332820300162333303162, −7.21149060870099590447476839146, −6.44567892308590497834738754983, −4.65020565674401390952975814410, −4.11159734733837588048764174943, −2.76607921556988363631501693627, −1.76786397007306889449089944138, 1.06813477039677088358281377118, 2.83449500444002418975213563928, 3.47339414534341283516784969024, 4.79148440330342991543316231734, 5.71955937431350302909481774793, 7.07548388300567085263041833922, 7.83687080150309299624298035045, 8.587073889579783411631015150150, 9.345383130371568112289591360070, 10.09449531567926161513314078553

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