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Properties

Label 2-72-8.5-c3-0-10
Degree $2$
Conductor $72$
Sign $0.940 + 0.339i$
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  + (2.80 + 0.325i)2-s + (7.78 + 1.83i)4-s − 18.5i·5-s + 9.32·7-s + (21.2 + 7.68i)8-s + (6.04 − 52.0i)10-s + 39.7i·11-s + 32.9i·13-s + (26.2 + 3.04i)14-s + (57.2 + 28.5i)16-s − 90.5·17-s − 72.5i·19-s + (33.9 − 144. i)20-s + (−12.9 + 111. i)22-s − 45.3·23-s + ⋯
L(s)  = 1  + (0.993 + 0.115i)2-s + (0.973 + 0.228i)4-s − 1.65i·5-s + 0.503·7-s + (0.940 + 0.339i)8-s + (0.191 − 1.64i)10-s + 1.08i·11-s + 0.703i·13-s + (0.500 + 0.0580i)14-s + (0.895 + 0.445i)16-s − 1.29·17-s − 0.876i·19-s + (0.379 − 1.61i)20-s + (−0.125 + 1.08i)22-s − 0.411·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.60417 - 0.455759i\)
\(L(\frac12)\) \(\approx\) \(2.60417 - 0.455759i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.80 - 0.325i)T \)
3 \( 1 \)
good5 \( 1 + 18.5iT - 125T^{2} \)
7 \( 1 - 9.32T + 343T^{2} \)
11 \( 1 - 39.7iT - 1.33e3T^{2} \)
13 \( 1 - 32.9iT - 2.19e3T^{2} \)
17 \( 1 + 90.5T + 4.91e3T^{2} \)
19 \( 1 + 72.5iT - 6.85e3T^{2} \)
23 \( 1 + 45.3T + 1.21e4T^{2} \)
29 \( 1 - 143. iT - 2.43e4T^{2} \)
31 \( 1 - 90.4T + 2.97e4T^{2} \)
37 \( 1 - 1.77iT - 5.06e4T^{2} \)
41 \( 1 + 195.T + 6.89e4T^{2} \)
43 \( 1 - 407. iT - 7.95e4T^{2} \)
47 \( 1 - 278.T + 1.03e5T^{2} \)
53 \( 1 + 241. iT - 1.48e5T^{2} \)
59 \( 1 - 149. iT - 2.05e5T^{2} \)
61 \( 1 + 508. iT - 2.26e5T^{2} \)
67 \( 1 + 950. iT - 3.00e5T^{2} \)
71 \( 1 - 803.T + 3.57e5T^{2} \)
73 \( 1 - 449.T + 3.89e5T^{2} \)
79 \( 1 + 157.T + 4.93e5T^{2} \)
83 \( 1 + 175. iT - 5.71e5T^{2} \)
89 \( 1 + 127.T + 7.04e5T^{2} \)
97 \( 1 - 158.T + 9.12e5T^{2} \)
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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

−13.88576627261405933574356304903, −12.95864787075364727770179031832, −12.19756373989451988137867418350, −11.16870265013637008373210004521, −9.399296279873317936535561392385, −8.190377517463941256120547409983, −6.73041038624901430387064650907, −4.98395312422848656000627081565, −4.42364770896817431671164723413, −1.83479725650786765735280718315, 2.50770158075500088906091389333, 3.81609962226723954690759502243, 5.74378298449712357146918945872, 6.75563828218632858940465637041, 8.045645377179463290449833295468, 10.29202774178244885596600016187, 10.98830124648593787073928448217, 11.85487696013106346625473796043, 13.45534945045106070492612877621, 14.13520928965667039850235907233

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