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Properties

Label 2-72-24.11-c3-0-3
Degree $2$
Conductor $72$
Sign $-0.515 - 0.856i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
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  + (1.35 + 2.48i)2-s + (−4.33 + 6.72i)4-s + 4.33·5-s + 28.4i·7-s + (−22.5 − 1.66i)8-s + (5.86 + 10.7i)10-s − 1.82i·11-s + 32.3i·13-s + (−70.5 + 38.4i)14-s + (−26.4 − 58.2i)16-s − 87.3i·17-s + 122.·19-s + (−18.7 + 29.1i)20-s + (4.53 − 2.47i)22-s + 68.6·23-s + ⋯
L(s)  = 1  + (0.478 + 0.878i)2-s + (−0.541 + 0.840i)4-s + 0.387·5-s + 1.53i·7-s + (−0.997 − 0.0735i)8-s + (0.185 + 0.340i)10-s − 0.0500i·11-s + 0.690i·13-s + (−1.34 + 0.734i)14-s + (−0.412 − 0.910i)16-s − 1.24i·17-s + 1.48·19-s + (−0.210 + 0.325i)20-s + (0.0439 − 0.0239i)22-s + 0.622·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.515 - 0.856i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.515 - 0.856i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.865494 + 1.53118i\)
\(L(\frac12)\) \(\approx\) \(0.865494 + 1.53118i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 2.48i)T \)
3 \( 1 \)
good5 \( 1 - 4.33T + 125T^{2} \)
7 \( 1 - 28.4iT - 343T^{2} \)
11 \( 1 + 1.82iT - 1.33e3T^{2} \)
13 \( 1 - 32.3iT - 2.19e3T^{2} \)
17 \( 1 + 87.3iT - 4.91e3T^{2} \)
19 \( 1 - 122.T + 6.85e3T^{2} \)
23 \( 1 - 68.6T + 1.21e4T^{2} \)
29 \( 1 - 297.T + 2.43e4T^{2} \)
31 \( 1 - 143. iT - 2.97e4T^{2} \)
37 \( 1 + 311. iT - 5.06e4T^{2} \)
41 \( 1 + 239. iT - 6.89e4T^{2} \)
43 \( 1 - 297.T + 7.95e4T^{2} \)
47 \( 1 + 311.T + 1.03e5T^{2} \)
53 \( 1 + 537.T + 1.48e5T^{2} \)
59 \( 1 - 661. iT - 2.05e5T^{2} \)
61 \( 1 - 69.7iT - 2.26e5T^{2} \)
67 \( 1 - 104.T + 3.00e5T^{2} \)
71 \( 1 + 601.T + 3.57e5T^{2} \)
73 \( 1 + 249.T + 3.89e5T^{2} \)
79 \( 1 - 151. iT - 4.93e5T^{2} \)
83 \( 1 + 515. iT - 5.71e5T^{2} \)
89 \( 1 - 507. iT - 7.04e5T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−14.41208983403646671004246847608, −13.73961425159860410384066773516, −12.35139422061836372233062484339, −11.66717964152961680465264988576, −9.510338158607421023027900975281, −8.777548822191472017714003840442, −7.27473591973563975410245373880, −5.93553578662177161712364296762, −4.95257154966238029030307249778, −2.84346342078865133803941704406, 1.12791016905174093749210906678, 3.31870713029144415817245916466, 4.71824097873668735931453414660, 6.28664879503063173225461186554, 7.962100586991455509919428186251, 9.758170940552203623287498901575, 10.40276832190872488545468879374, 11.49110308685424521928962496835, 12.85983551052489750742978882086, 13.61677610425414834411986080019

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