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Properties

Label 2-72-72.43-c2-0-21
Degree $2$
Conductor $72$
Sign $-0.754 + 0.656i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
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  + (1 − 1.73i)2-s + (−1.94 − 2.28i)3-s + (−1.99 − 3.46i)4-s + (−5.89 + 1.09i)6-s − 7.99·8-s + (−1.39 + 8.89i)9-s + (10.8 − 18.7i)11-s + (−4 + 11.3i)12-s + (−8 + 13.8i)16-s + 28.3·17-s + (14 + 11.3i)18-s + 2.30·19-s + (−21.6 − 37.5i)22-s + (15.5 + 18.2i)24-s + (−12.5 + 21.6i)25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.649 − 0.760i)3-s + (−0.499 − 0.866i)4-s + (−0.983 + 0.182i)6-s − 0.999·8-s + (−0.155 + 0.987i)9-s + (0.986 − 1.70i)11-s + (−0.333 + 0.942i)12-s + (−0.5 + 0.866i)16-s + 1.67·17-s + (0.777 + 0.628i)18-s + 0.121·19-s + (−0.986 − 1.70i)22-s + (0.649 + 0.760i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.754 + 0.656i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ -0.754 + 0.656i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.425825 - 1.13717i\)
\(L(\frac12)\) \(\approx\) \(0.425825 - 1.13717i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (1.94 + 2.28i)T \)
good5 \( 1 + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.8 + 18.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (84.5 - 146. i)T^{2} \)
17 \( 1 - 28.3T + 289T^{2} \)
19 \( 1 - 2.30T + 361T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + (-17.8 - 30.9i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (33.2 - 57.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (57.2 + 99.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-66.9 - 115. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 100.T + 5.32e3T^{2} \)
79 \( 1 + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (79 - 136. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 146T + 7.92e3T^{2} \)
97 \( 1 + (-49.9 + 86.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.77594598014821856718337582805, −12.76746412488851413215894776888, −11.67820099464043909165262910537, −11.13564107554154858810613089976, −9.707025863920982988366364925845, −8.183835410055866268691537998988, −6.34820666215631933030829094592, −5.38294737984831968263409884575, −3.39335713498662159480546664509, −1.14658148866337258030468862217, 3.82871892362638352406689144689, 4.98888721332793020568271603921, 6.28444123948809859784341025928, 7.51975771124509058498275340277, 9.207086370509510394378466322293, 10.13211271349233195653081717861, 11.96491239319616093968301261848, 12.37911517367329109183788698111, 14.14016535253452506721659009985, 14.90403468040892537349100709675

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