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Properties

Label 2-72-72.11-c3-0-12
Degree $2$
Conductor $72$
Sign $0.420 - 0.907i$
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  + (−1.47 + 2.41i)2-s + (5.12 − 0.875i)3-s + (−3.64 − 7.11i)4-s + (3.60 + 6.24i)5-s + (−5.44 + 13.6i)6-s + (6.44 + 3.72i)7-s + (22.5 + 1.69i)8-s + (25.4 − 8.97i)9-s + (−20.4 − 0.510i)10-s + (35.6 + 20.5i)11-s + (−24.9 − 33.2i)12-s + (−26.5 + 15.3i)13-s + (−18.5 + 10.0i)14-s + (23.9 + 28.8i)15-s + (−37.3 + 51.9i)16-s + 46.2i·17-s + ⋯
L(s)  = 1  + (−0.521 + 0.853i)2-s + (0.985 − 0.168i)3-s + (−0.456 − 0.889i)4-s + (0.322 + 0.558i)5-s + (−0.370 + 0.928i)6-s + (0.348 + 0.201i)7-s + (0.997 + 0.0749i)8-s + (0.943 − 0.332i)9-s + (−0.645 − 0.0161i)10-s + (0.977 + 0.564i)11-s + (−0.599 − 0.800i)12-s + (−0.566 + 0.327i)13-s + (−0.353 + 0.192i)14-s + (0.412 + 0.496i)15-s + (−0.583 + 0.811i)16-s + 0.659i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.420 - 0.907i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.37473 + 0.877665i\)
\(L(\frac12)\) \(\approx\) \(1.37473 + 0.877665i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.47 - 2.41i)T \)
3 \( 1 + (-5.12 + 0.875i)T \)
good5 \( 1 + (-3.60 - 6.24i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-6.44 - 3.72i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-35.6 - 20.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (26.5 - 15.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 46.2iT - 4.91e3T^{2} \)
19 \( 1 + 31.7T + 6.85e3T^{2} \)
23 \( 1 + (25.4 + 44.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-135. + 234. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (226. - 130. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 372. iT - 5.06e4T^{2} \)
41 \( 1 + (159. - 91.8i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-41.4 + 71.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (141. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 442.T + 1.48e5T^{2} \)
59 \( 1 + (563. - 325. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (44.7 + 25.8i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (360. + 623. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 794.T + 3.57e5T^{2} \)
73 \( 1 + 639.T + 3.89e5T^{2} \)
79 \( 1 + (-901. - 520. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (80.8 + 46.6i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.27e3iT - 7.04e5T^{2} \)
97 \( 1 + (-181. + 314. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.62188343164820605066434314636, −13.77352660744667635419453875961, −12.33375928367409332716684051350, −10.52809247846080434101867340827, −9.505952248036107469330787116166, −8.541860361330031749023220309789, −7.32646315016493791030986611338, −6.32892433671316700605465567449, −4.33055511806290096960641817129, −1.96600667225499185095085955346, 1.48191464641600841130980792101, 3.25615479549158166976864309305, 4.74250464596994374271079064157, 7.29794568911162416777825040543, 8.581312396851243417833172780608, 9.278345441751218919850475326466, 10.36833112166666863896479578055, 11.69164683813865324574449836559, 12.88114210775428819393391572207, 13.76905412551933317345626169699

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