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Properties

Label 2-24e2-9.2-c2-0-1
Degree 22
Conductor 576576
Sign −0.452−0.891i-0.452 - 0.891i
Analytic cond. 15.694815.6948
Root an. cond. 3.961673.96167
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.65 − 2.50i)3-s + (0.721 − 0.416i)5-s + (1.26 − 2.18i)7-s + (−3.53 + 8.27i)9-s + (−9.47 − 5.47i)11-s + (−4.36 − 7.55i)13-s + (−2.23 − 1.11i)15-s + 20.8i·17-s − 1.50·19-s + (−7.54 + 0.454i)21-s + (−1.00 + 0.578i)23-s + (−12.1 + 21.0i)25-s + (26.5 − 4.84i)27-s + (−15.7 − 9.08i)29-s + (−25.6 − 44.4i)31-s + ⋯
L(s)  = 1  + (−0.551 − 0.834i)3-s + (0.144 − 0.0832i)5-s + (0.180 − 0.311i)7-s + (−0.392 + 0.919i)9-s + (−0.861 − 0.497i)11-s + (−0.335 − 0.581i)13-s + (−0.148 − 0.0744i)15-s + 1.22i·17-s − 0.0790·19-s + (−0.359 + 0.0216i)21-s + (−0.0435 + 0.0251i)23-s + (−0.486 + 0.842i)25-s + (0.983 − 0.179i)27-s + (−0.542 − 0.313i)29-s + (−0.828 − 1.43i)31-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=((−0.452−0.891i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+1) L(s)=((−0.452−0.891i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: −0.452−0.891i-0.452 - 0.891i
Analytic conductor: 15.694815.6948
Root analytic conductor: 3.961673.96167
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ576(65,⋅)\chi_{576} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :1), −0.452−0.891i)(2,\ 576,\ (\ :1),\ -0.452 - 0.891i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.14204115360.1420411536
L(12)L(\frac12) ≈\approx 0.14204115360.1420411536
L(2)L(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
3 1+(1.65+2.50i)T 1 + (1.65 + 2.50i)T
good5 1+(−0.721+0.416i)T+(12.5−21.6i)T2 1 + (-0.721 + 0.416i)T + (12.5 - 21.6i)T^{2}
7 1+(−1.26+2.18i)T+(−24.5−42.4i)T2 1 + (-1.26 + 2.18i)T + (-24.5 - 42.4i)T^{2}
11 1+(9.47+5.47i)T+(60.5+104.i)T2 1 + (9.47 + 5.47i)T + (60.5 + 104. i)T^{2}
13 1+(4.36+7.55i)T+(−84.5+146.i)T2 1 + (4.36 + 7.55i)T + (-84.5 + 146. i)T^{2}
17 1−20.8iT−289T2 1 - 20.8iT - 289T^{2}
19 1+1.50T+361T2 1 + 1.50T + 361T^{2}
23 1+(1.00−0.578i)T+(264.5−458.i)T2 1 + (1.00 - 0.578i)T + (264.5 - 458. i)T^{2}
29 1+(15.7+9.08i)T+(420.5+728.i)T2 1 + (15.7 + 9.08i)T + (420.5 + 728. i)T^{2}
31 1+(25.6+44.4i)T+(−480.5+832.i)T2 1 + (25.6 + 44.4i)T + (-480.5 + 832. i)T^{2}
37 1−7.93T+1.36e3T2 1 - 7.93T + 1.36e3T^{2}
41 1+(21.8−12.6i)T+(840.5−1.45e3i)T2 1 + (21.8 - 12.6i)T + (840.5 - 1.45e3i)T^{2}
43 1+(19.3−33.5i)T+(−924.5−1.60e3i)T2 1 + (19.3 - 33.5i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−59.6−34.4i)T+(1.10e3+1.91e3i)T2 1 + (-59.6 - 34.4i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−46.5iT−2.80e3T2 1 - 46.5iT - 2.80e3T^{2}
59 1+(89.1−51.4i)T+(1.74e3−3.01e3i)T2 1 + (89.1 - 51.4i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(44.1−76.4i)T+(−1.86e3−3.22e3i)T2 1 + (44.1 - 76.4i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−11.3−19.6i)T+(−2.24e3+3.88e3i)T2 1 + (-11.3 - 19.6i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1−104.iT−5.04e3T2 1 - 104. iT - 5.04e3T^{2}
73 1+75.2T+5.32e3T2 1 + 75.2T + 5.32e3T^{2}
79 1+(−51.8+89.8i)T+(−3.12e3−5.40e3i)T2 1 + (-51.8 + 89.8i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(53.7+31.0i)T+(3.44e3+5.96e3i)T2 1 + (53.7 + 31.0i)T + (3.44e3 + 5.96e3i)T^{2}
89 1−1.95iT−7.92e3T2 1 - 1.95iT - 7.92e3T^{2}
97 1+(−59.2+102.i)T+(−4.70e3−8.14e3i)T2 1 + (-59.2 + 102. i)T + (-4.70e3 - 8.14e3i)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

−10.85733369222221715752944393192, −10.19504846852235187081344063074, −8.920179527688782670562715014204, −7.78883779330419789963098271385, −7.50005355098402734638213927746, −6.00325392531820015225026556869, −5.63563136076709397318296211579, −4.29852454790744554785824223446, −2.76648720993658081120129798856, −1.45674825101403057450814225315, 0.05671497052860884516125591841, 2.20556674324524352417737642225, 3.53471684823887757337437459670, 4.84517637532556337948418045441, 5.30012726604039642451968128257, 6.53670025532777651554678777260, 7.45160377047705106175440696297, 8.719963498606355510801968166803, 9.474803956299502163672271152381, 10.26668110094478457831271369784

Graph of the ZZ-function along the critical line