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Properties

Label 2-720-9.7-c1-0-5
Degree $2$
Conductor $720$
Sign $-0.283 - 0.959i$
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  + (−0.5 + 1.65i)3-s + (0.5 + 0.866i)5-s + (1.68 − 2.92i)7-s + (−2.5 − 1.65i)9-s + (−2.18 + 3.78i)11-s + (3.37 + 5.84i)13-s + (−1.68 + 0.396i)15-s − 1.62·17-s + 2.37·19-s + (4 + 4.25i)21-s + (0.686 + 1.18i)23-s + (−0.499 + 0.866i)25-s + (4 − 3.31i)27-s + (−0.686 + 1.18i)29-s + (2.37 + 4.10i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.957i)3-s + (0.223 + 0.387i)5-s + (0.637 − 1.10i)7-s + (−0.833 − 0.552i)9-s + (−0.659 + 1.14i)11-s + (0.935 + 1.61i)13-s + (−0.435 + 0.102i)15-s − 0.394·17-s + 0.544·19-s + (0.872 + 0.928i)21-s + (0.143 + 0.247i)23-s + (−0.0999 + 0.173i)25-s + (0.769 − 0.638i)27-s + (−0.127 + 0.220i)29-s + (0.426 + 0.737i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.283 - 0.959i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.283 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799200 + 1.06914i\)
\(L(\frac12)\) \(\approx\) \(0.799200 + 1.06914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.68 + 2.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.37 - 5.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (-0.686 - 1.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.686 - 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.37 - 4.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.81 - 4.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.68 - 6.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (-2.18 - 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 3.11T + 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.68 + 6.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76798313301803499142644018688, −9.851211446780944221842307233630, −9.230408248345664959414744372672, −8.092013707512517075329414430753, −7.07029882326073477280732772888, −6.31064795445806585845113464698, −4.90441204968841974734233093047, −4.42012588764006285357093255135, −3.33608604005334151096161475763, −1.67968990075918061043865538968, 0.74678151871686385370391301464, 2.19985562581886842298166770815, 3.25853268780041647815861755521, 5.24934320231587197705396059503, 5.55820279733234683070623559558, 6.42181100335918809527847325685, 7.86143608382591532912354156359, 8.296947592819329347751373051126, 8.939802449387705163626122805743, 10.38237882523974941432845660824

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