Properties

Label 2-432-27.4-c1-0-7
Degree 22
Conductor 432432
Sign 0.7430.669i0.743 - 0.669i
Analytic cond. 3.449533.44953
Root an. cond. 1.857291.85729
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.45 + 0.940i)3-s + (−0.196 + 1.11i)5-s + (2.99 − 2.51i)7-s + (1.23 − 2.73i)9-s + (0.324 + 1.84i)11-s + (0.688 − 0.250i)13-s + (−0.760 − 1.80i)15-s + (−0.944 + 1.63i)17-s + (1.37 + 2.37i)19-s + (−1.99 + 6.47i)21-s + (4.46 + 3.74i)23-s + (3.49 + 1.27i)25-s + (0.782 + 5.13i)27-s + (4.99 + 1.81i)29-s + (−1.02 − 0.861i)31-s + ⋯
L(s)  = 1  + (−0.839 + 0.542i)3-s + (−0.0877 + 0.497i)5-s + (1.13 − 0.949i)7-s + (0.410 − 0.911i)9-s + (0.0979 + 0.555i)11-s + (0.190 − 0.0694i)13-s + (−0.196 − 0.465i)15-s + (−0.229 + 0.396i)17-s + (0.314 + 0.544i)19-s + (−0.434 + 1.41i)21-s + (0.930 + 0.781i)23-s + (0.699 + 0.254i)25-s + (0.150 + 0.988i)27-s + (0.928 + 0.337i)29-s + (−0.184 − 0.154i)31-s + ⋯

Functional equation

Λ(s)=(432s/2ΓC(s)L(s)=((0.7430.669i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(432s/2ΓC(s+1/2)L(s)=((0.7430.669i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 432432    =    24332^{4} \cdot 3^{3}
Sign: 0.7430.669i0.743 - 0.669i
Analytic conductor: 3.449533.44953
Root analytic conductor: 1.857291.85729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ432(193,)\chi_{432} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 432, ( :1/2), 0.7430.669i)(2,\ 432,\ (\ :1/2),\ 0.743 - 0.669i)

Particular Values

L(1)L(1) \approx 1.11968+0.429776i1.11968 + 0.429776i
L(12)L(\frac12) \approx 1.11968+0.429776i1.11968 + 0.429776i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.450.940i)T 1 + (1.45 - 0.940i)T
good5 1+(0.1961.11i)T+(4.691.71i)T2 1 + (0.196 - 1.11i)T + (-4.69 - 1.71i)T^{2}
7 1+(2.99+2.51i)T+(1.216.89i)T2 1 + (-2.99 + 2.51i)T + (1.21 - 6.89i)T^{2}
11 1+(0.3241.84i)T+(10.3+3.76i)T2 1 + (-0.324 - 1.84i)T + (-10.3 + 3.76i)T^{2}
13 1+(0.688+0.250i)T+(9.958.35i)T2 1 + (-0.688 + 0.250i)T + (9.95 - 8.35i)T^{2}
17 1+(0.9441.63i)T+(8.514.7i)T2 1 + (0.944 - 1.63i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.372.37i)T+(9.5+16.4i)T2 1 + (-1.37 - 2.37i)T + (-9.5 + 16.4i)T^{2}
23 1+(4.463.74i)T+(3.99+22.6i)T2 1 + (-4.46 - 3.74i)T + (3.99 + 22.6i)T^{2}
29 1+(4.991.81i)T+(22.2+18.6i)T2 1 + (-4.99 - 1.81i)T + (22.2 + 18.6i)T^{2}
31 1+(1.02+0.861i)T+(5.38+30.5i)T2 1 + (1.02 + 0.861i)T + (5.38 + 30.5i)T^{2}
37 1+(1.692.94i)T+(18.532.0i)T2 1 + (1.69 - 2.94i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.680.614i)T+(31.426.3i)T2 1 + (1.68 - 0.614i)T + (31.4 - 26.3i)T^{2}
43 1+(0.873+4.95i)T+(40.4+14.7i)T2 1 + (0.873 + 4.95i)T + (-40.4 + 14.7i)T^{2}
47 1+(1.301.09i)T+(8.1646.2i)T2 1 + (1.30 - 1.09i)T + (8.16 - 46.2i)T^{2}
53 12.84T+53T2 1 - 2.84T + 53T^{2}
59 1+(1.95+11.0i)T+(55.420.1i)T2 1 + (-1.95 + 11.0i)T + (-55.4 - 20.1i)T^{2}
61 1+(4.00+3.36i)T+(10.560.0i)T2 1 + (-4.00 + 3.36i)T + (10.5 - 60.0i)T^{2}
67 1+(1.770.646i)T+(51.343.0i)T2 1 + (1.77 - 0.646i)T + (51.3 - 43.0i)T^{2}
71 1+(6.0910.5i)T+(35.561.4i)T2 1 + (6.09 - 10.5i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.94+8.56i)T+(36.5+63.2i)T2 1 + (4.94 + 8.56i)T + (-36.5 + 63.2i)T^{2}
79 1+(11.64.22i)T+(60.5+50.7i)T2 1 + (-11.6 - 4.22i)T + (60.5 + 50.7i)T^{2}
83 1+(10.9+3.99i)T+(63.5+53.3i)T2 1 + (10.9 + 3.99i)T + (63.5 + 53.3i)T^{2}
89 1+(2.864.96i)T+(44.5+77.0i)T2 1 + (-2.86 - 4.96i)T + (-44.5 + 77.0i)T^{2}
97 1+(0.0596+0.338i)T+(91.1+33.1i)T2 1 + (0.0596 + 0.338i)T + (-91.1 + 33.1i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.10047041941858806271154347718, −10.56376032892829431162107675645, −9.763945750588420521850004565172, −8.519202639335075244826865578614, −7.36199570764663661004537464415, −6.68039515523202993660838598976, −5.33621891069289972892059022064, −4.53287962754311816452090860443, −3.48585001159102816200903703958, −1.35491781042669014145476312652, 1.06976758893814698218807919890, 2.55182322370590435894195993184, 4.63368356682946590020536070439, 5.21844234629027587889706846666, 6.24442319537071030236421314043, 7.30141640757458762335848476501, 8.432431483713410146021008407528, 8.919597912690041725009589347154, 10.43903444338181741013050161255, 11.30238050615807118963712614371

Graph of the ZZ-function along the critical line