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Properties

Label 2-72-8.3-c6-0-18
Degree $2$
Conductor $72$
Sign $-0.994 + 0.106i$
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  + (−4.24 − 6.78i)2-s + (−27.9 + 57.5i)4-s + 206. i·5-s + 210. i·7-s + (509. − 54.6i)8-s + (1.39e3 − 874. i)10-s − 2.26e3·11-s − 3.53e3i·13-s + (1.42e3 − 893. i)14-s + (−2.53e3 − 3.22e3i)16-s − 3.48e3·17-s + 9.84e3·19-s + (−1.18e4 − 5.76e3i)20-s + (9.60e3 + 1.53e4i)22-s − 2.09e3i·23-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)2-s + (−0.437 + 0.899i)4-s + 1.64i·5-s + 0.613i·7-s + (0.994 − 0.106i)8-s + (1.39 − 0.874i)10-s − 1.69·11-s − 1.61i·13-s + (0.519 − 0.325i)14-s + (−0.618 − 0.786i)16-s − 0.709·17-s + 1.43·19-s + (−1.48 − 0.720i)20-s + (0.901 + 1.44i)22-s − 0.172i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.994 + 0.106i$
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: \((2,\ 72,\ (\ :3),\ -0.994 + 0.106i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00105154 - 0.0196388i\)
\(L(\frac12)\) \(\approx\) \(0.00105154 - 0.0196388i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.24 + 6.78i)T \)
3 \( 1 \)
good5 \( 1 - 206. iT - 1.56e4T^{2} \)
7 \( 1 - 210. iT - 1.17e5T^{2} \)
11 \( 1 + 2.26e3T + 1.77e6T^{2} \)
13 \( 1 + 3.53e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.48e3T + 2.41e7T^{2} \)
19 \( 1 - 9.84e3T + 4.70e7T^{2} \)
23 \( 1 + 2.09e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.01e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.37e3iT - 8.87e8T^{2} \)
37 \( 1 + 5.50e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.84e4T + 4.75e9T^{2} \)
43 \( 1 + 1.06e5T + 6.32e9T^{2} \)
47 \( 1 - 8.40e4iT - 1.07e10T^{2} \)
53 \( 1 + 8.03e4iT - 2.21e10T^{2} \)
59 \( 1 - 8.45e4T + 4.21e10T^{2} \)
61 \( 1 + 1.60e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.29e5T + 9.04e10T^{2} \)
71 \( 1 - 1.84e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.56e5T + 1.51e11T^{2} \)
79 \( 1 - 4.43e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.56e5T + 3.26e11T^{2} \)
89 \( 1 + 5.93e5T + 4.96e11T^{2} \)
97 \( 1 + 7.83e5T + 8.32e11T^{2} \)
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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

−12.78657608689826282433197813088, −11.44623650232321451880495594140, −10.58621491076678499246188846292, −9.912779343604545593543005575602, −8.188857105854469365677828676967, −7.28841140158708465692282381958, −5.44944987021826412827234584809, −3.17677121877756230554561224997, −2.50230032451856991397356756298, −0.009175411607964685125185337746, 1.44755788734617998071794347219, 4.53574621112739838715472067101, 5.35372436679405032611323995695, 7.08494436098505693599911609139, 8.229884137109328429339944641179, 9.154159681357433440372002012774, 10.20902919440222369114861876401, 11.74117977059344868546071632740, 13.31552286974823397541542086481, 13.69758493633853789496969377249

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