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Properties

Label 2-72-8.3-c6-0-24
Degree $2$
Conductor $72$
Sign $0.474 + 0.880i$
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  + (7.49 + 2.80i)2-s + (48.2 + 42.0i)4-s − 128. i·5-s − 534. i·7-s + (242. + 450. i)8-s + (360. − 961. i)10-s − 1.81e3·11-s − 3.11e3i·13-s + (1.50e3 − 4.00e3i)14-s + (552. + 4.05e3i)16-s + 1.24e3·17-s + 9.24e3·19-s + (5.40e3 − 6.18e3i)20-s + (−1.35e4 − 5.09e3i)22-s − 403. i·23-s + ⋯
L(s)  = 1  + (0.936 + 0.351i)2-s + (0.753 + 0.657i)4-s − 1.02i·5-s − 1.55i·7-s + (0.474 + 0.880i)8-s + (0.360 − 0.961i)10-s − 1.36·11-s − 1.41i·13-s + (0.547 − 1.45i)14-s + (0.134 + 0.990i)16-s + 0.253·17-s + 1.34·19-s + (0.675 − 0.773i)20-s + (−1.27 − 0.478i)22-s − 0.0331i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(16.5638\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3),\ 0.474 + 0.880i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.51762 - 1.50328i\)
\(L(\frac12)\) \(\approx\) \(2.51762 - 1.50328i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.49 - 2.80i)T \)
3 \( 1 \)
good5 \( 1 + 128. iT - 1.56e4T^{2} \)
7 \( 1 + 534. iT - 1.17e5T^{2} \)
11 \( 1 + 1.81e3T + 1.77e6T^{2} \)
13 \( 1 + 3.11e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.24e3T + 2.41e7T^{2} \)
19 \( 1 - 9.24e3T + 4.70e7T^{2} \)
23 \( 1 + 403. iT - 1.48e8T^{2} \)
29 \( 1 - 1.32e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.40e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.83e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.04e4T + 4.75e9T^{2} \)
43 \( 1 + 9.92e3T + 6.32e9T^{2} \)
47 \( 1 - 9.96e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.15e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.73e5T + 4.21e10T^{2} \)
61 \( 1 + 1.44e4iT - 5.15e10T^{2} \)
67 \( 1 + 3.11e5T + 9.04e10T^{2} \)
71 \( 1 - 4.92e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.53e5T + 1.51e11T^{2} \)
79 \( 1 - 3.34e5iT - 2.43e11T^{2} \)
83 \( 1 + 4.18e5T + 3.26e11T^{2} \)
89 \( 1 - 1.97e5T + 4.96e11T^{2} \)
97 \( 1 - 1.30e6T + 8.32e11T^{2} \)
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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

−13.17410872727737813237946796195, −12.65598165566179047880669981709, −11.10696309843897329520309451145, −10.06413283211809152340429158408, −8.057865421295050510544995381544, −7.43092868745335870904412586070, −5.58368261408531800964660722159, −4.64452719842063864678636851996, −3.16773694224291497437019605472, −0.842221231888670583231315566789, 2.17330620670732532064762311304, 3.12631347850726643634881365939, 5.05703960984728754724775694543, 6.13240791001981508789901289797, 7.42621673329644844536838923401, 9.276061247726038638862386551179, 10.55750488780767042949419363809, 11.59507509603364686104792920784, 12.37831361852928667724433326925, 13.71184154234535400907166741117

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