Properties

Label 2-72-9.5-c6-0-13
Degree $2$
Conductor $72$
Sign $0.348 + 0.937i$
Analytic cond. $16.5638$
Root an. cond. $4.06987$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (22.0 − 15.5i)3-s + (−91.6 − 52.9i)5-s + (238. + 412. i)7-s + (244. − 686. i)9-s + (1.89e3 − 1.09e3i)11-s + (353. − 611. i)13-s + (−2.84e3 + 259. i)15-s − 8.18e3i·17-s − 7.00e3·19-s + (1.16e4 + 5.39e3i)21-s + (6.76e3 + 3.90e3i)23-s + (−2.20e3 − 3.82e3i)25-s + (−5.30e3 − 1.89e4i)27-s + (−1.63e4 + 9.44e3i)29-s + (2.46e4 − 4.26e4i)31-s + ⋯
L(s)  = 1  + (0.817 − 0.576i)3-s + (−0.733 − 0.423i)5-s + (0.695 + 1.20i)7-s + (0.335 − 0.942i)9-s + (1.42 − 0.824i)11-s + (0.160 − 0.278i)13-s + (−0.843 + 0.0769i)15-s − 1.66i·17-s − 1.02·19-s + (1.26 + 0.582i)21-s + (0.556 + 0.321i)23-s + (−0.141 − 0.244i)25-s + (−0.269 − 0.963i)27-s + (−0.670 + 0.387i)29-s + (0.826 − 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.94845 - 1.35368i\)
\(L(\frac12)\) \(\approx\) \(1.94845 - 1.35368i\)
\(L(4)\) 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 \( 1 + (-22.0 + 15.5i)T \)
good5 \( 1 + (91.6 + 52.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-238. - 412. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-1.89e3 + 1.09e3i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (-353. + 611. i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + 8.18e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.00e3T + 4.70e7T^{2} \)
23 \( 1 + (-6.76e3 - 3.90e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (1.63e4 - 9.44e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-2.46e4 + 4.26e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 1.29e3T + 2.56e9T^{2} \)
41 \( 1 + (-8.20e4 - 4.73e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (-2.80e4 - 4.85e4i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-9.63e4 + 5.56e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 - 1.62e5iT - 2.21e10T^{2} \)
59 \( 1 + (2.37e4 + 1.37e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.03e5 - 1.79e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.90e5 - 5.03e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 1.23e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.01e5T + 1.51e11T^{2} \)
79 \( 1 + (3.48e5 + 6.04e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.30e5 + 1.90e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 3.50e5iT - 4.96e11T^{2} \)
97 \( 1 + (-4.42e5 - 7.66e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−13.22221173539901295924956622088, −11.95373407825774034088523255755, −11.50480674615352632426922945567, −9.211621853032338593898476566104, −8.663229597279987364913511502574, −7.55922934695285418183404093942, −6.02624639817681746931046582924, −4.22677673986677766278587740119, −2.63012122307788854211807639444, −0.939734437728600147310874871563, 1.64846128983186972456215405433, 3.82111798897530471086183004964, 4.32479061409010887143700098313, 6.80622607934080225765600686502, 7.85747741053807639850383283284, 8.973561722755553118461798811695, 10.39733258332008763982747528077, 11.10374162852443757707774631575, 12.58670682406688460519852742016, 14.00931411408765030278501160053

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