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Properties

Label 2-72-72.11-c3-0-12
Degree 22
Conductor 7272
Sign 0.420−0.907i0.420 - 0.907i
Analytic cond. 4.248134.24813
Root an. cond. 2.061102.06110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(72s/2 ΓC(s) L(s)=((0.420−0.907i) Λ‾(4−s)\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}
Λ(s)=(72s/2 ΓC(s+3/2) L(s)=((0.420−0.907i) Λ‾(1−s)\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: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.420−0.907i0.420 - 0.907i
Analytic conductor: 4.248134.24813
Root analytic conductor: 2.061102.06110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ72(11,⋅)\chi_{72} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :3/2), 0.420−0.907i)(2,\ 72,\ (\ :3/2),\ 0.420 - 0.907i)

Particular Values

L(2)L(2) ≈\approx 1.37473+0.877665i1.37473 + 0.877665i
L(12)L(\frac12) ≈\approx 1.37473+0.877665i1.37473 + 0.877665i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.47−2.41i)T 1 + (1.47 - 2.41i)T
3 1+(−5.12+0.875i)T 1 + (-5.12 + 0.875i)T
good5 1+(−3.60−6.24i)T+(−62.5+108.i)T2 1 + (-3.60 - 6.24i)T + (-62.5 + 108. i)T^{2}
7 1+(−6.44−3.72i)T+(171.5+297.i)T2 1 + (-6.44 - 3.72i)T + (171.5 + 297. i)T^{2}
11 1+(−35.6−20.5i)T+(665.5+1.15e3i)T2 1 + (-35.6 - 20.5i)T + (665.5 + 1.15e3i)T^{2}
13 1+(26.5−15.3i)T+(1.09e3−1.90e3i)T2 1 + (26.5 - 15.3i)T + (1.09e3 - 1.90e3i)T^{2}
17 1−46.2iT−4.91e3T2 1 - 46.2iT - 4.91e3T^{2}
19 1+31.7T+6.85e3T2 1 + 31.7T + 6.85e3T^{2}
23 1+(25.4+44.1i)T+(−6.08e3+1.05e4i)T2 1 + (25.4 + 44.1i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−135.+234.i)T+(−1.21e4−2.11e4i)T2 1 + (-135. + 234. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(226.−130.i)T+(1.48e4−2.57e4i)T2 1 + (226. - 130. i)T + (1.48e4 - 2.57e4i)T^{2}
37 1+372.iT−5.06e4T2 1 + 372. iT - 5.06e4T^{2}
41 1+(159.−91.8i)T+(3.44e4−5.96e4i)T2 1 + (159. - 91.8i)T + (3.44e4 - 5.96e4i)T^{2}
43 1+(−41.4+71.8i)T+(−3.97e4−6.88e4i)T2 1 + (-41.4 + 71.8i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(141.−244.i)T+(−5.19e4−8.99e4i)T2 1 + (141. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−442.T+1.48e5T2 1 - 442.T + 1.48e5T^{2}
59 1+(563.−325.i)T+(1.02e5−1.77e5i)T2 1 + (563. - 325. i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(44.7+25.8i)T+(1.13e5+1.96e5i)T2 1 + (44.7 + 25.8i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(360.+623.i)T+(−1.50e5+2.60e5i)T2 1 + (360. + 623. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+794.T+3.57e5T2 1 + 794.T + 3.57e5T^{2}
73 1+639.T+3.89e5T2 1 + 639.T + 3.89e5T^{2}
79 1+(−901.−520.i)T+(2.46e5+4.26e5i)T2 1 + (-901. - 520. i)T + (2.46e5 + 4.26e5i)T^{2}
83 1+(80.8+46.6i)T+(2.85e5+4.95e5i)T2 1 + (80.8 + 46.6i)T + (2.85e5 + 4.95e5i)T^{2}
89 1−1.27e3iT−7.04e5T2 1 - 1.27e3iT - 7.04e5T^{2}
97 1+(−181.+314.i)T+(−4.56e5−7.90e5i)T2 1 + (-181. + 314. i)T + (-4.56e5 - 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line