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Properties

Label 2-72-9.4-c3-0-5
Degree $2$
Conductor $72$
Sign $-0.0353 + 0.999i$
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  + (−4.74 + 2.11i)3-s + (2.99 − 5.19i)5-s + (−7.78 − 13.4i)7-s + (18.0 − 20.0i)9-s + (−17.4 − 30.2i)11-s + (40.3 − 69.8i)13-s + (−3.26 + 30.9i)15-s − 70.1·17-s − 4.25·19-s + (65.4 + 47.5i)21-s + (−59.3 + 102. i)23-s + (44.5 + 77.1i)25-s + (−43.4 + 133. i)27-s + (−61.7 − 106. i)29-s + (92.8 − 160. i)31-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (0.268 − 0.464i)5-s + (−0.420 − 0.728i)7-s + (0.669 − 0.742i)9-s + (−0.478 − 0.828i)11-s + (0.860 − 1.48i)13-s + (−0.0561 + 0.533i)15-s − 1.00·17-s − 0.0513·19-s + (0.680 + 0.494i)21-s + (−0.538 + 0.932i)23-s + (0.356 + 0.617i)25-s + (−0.309 + 0.950i)27-s + (−0.395 − 0.684i)29-s + (0.538 − 0.931i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0353 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0353 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.0353 + 0.999i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.0353 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.587898 - 0.609039i\)
\(L(\frac12)\) \(\approx\) \(0.587898 - 0.609039i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.74 - 2.11i)T \)
good5 \( 1 + (-2.99 + 5.19i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (7.78 + 13.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (17.4 + 30.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-40.3 + 69.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 70.1T + 4.91e3T^{2} \)
19 \( 1 + 4.25T + 6.85e3T^{2} \)
23 \( 1 + (59.3 - 102. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (61.7 + 106. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-92.8 + 160. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 151.T + 5.06e4T^{2} \)
41 \( 1 + (106. - 183. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (145. + 251. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-106. - 184. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 556.T + 1.48e5T^{2} \)
59 \( 1 + (-426. + 738. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-344. - 596. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-457. + 793. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 786.T + 3.57e5T^{2} \)
73 \( 1 + 993.T + 3.89e5T^{2} \)
79 \( 1 + (-284. - 492. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-373. - 647. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + (-609. - 1.05e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.41310451552486820315366024415, −13.01837355813282887715392825192, −11.40747409306965611261470936111, −10.60383367072305830503751340349, −9.544692241891875659913042015658, −8.017146405882355566503028123052, −6.33834133014692454031860336907, −5.30919338371228823416478292194, −3.68683713253948611599322517690, −0.61589589470917564577132320657, 2.13674546590426528633698878599, 4.60412771102391611805970913123, 6.20834565079362377151858301250, 6.91506927913438253336141816105, 8.718667627042022787784633652095, 10.11440708973610179369848394653, 11.17904314153838889093427936612, 12.21021258604384739233764329913, 13.13982119229295774104416192341, 14.32824102669677498449959751542

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