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Properties

Label 2-432-27.16-c1-0-16
Degree 22
Conductor 432432
Sign −0.953−0.299i-0.953 - 0.299i
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  + (−0.159 − 1.72i)3-s + (−2.08 − 0.757i)5-s + (0.229 − 1.29i)7-s + (−2.94 + 0.551i)9-s + (−4.90 + 1.78i)11-s + (−0.0138 + 0.0116i)13-s + (−0.974 + 3.71i)15-s + (1.56 + 2.71i)17-s + (0.208 − 0.361i)19-s + (−2.27 − 0.187i)21-s + (0.179 + 1.01i)23-s + (−0.0712 − 0.0597i)25-s + (1.42 + 4.99i)27-s + (−5.98 − 5.01i)29-s + (−0.647 − 3.67i)31-s + ⋯
L(s)  = 1  + (−0.0922 − 0.995i)3-s + (−0.930 − 0.338i)5-s + (0.0866 − 0.491i)7-s + (−0.982 + 0.183i)9-s + (−1.47 + 0.537i)11-s + (−0.00383 + 0.00321i)13-s + (−0.251 + 0.958i)15-s + (0.379 + 0.658i)17-s + (0.0478 − 0.0829i)19-s + (−0.497 − 0.0409i)21-s + (0.0374 + 0.212i)23-s + (−0.0142 − 0.0119i)25-s + (0.273 + 0.961i)27-s + (−1.11 − 0.931i)29-s + (−0.116 − 0.659i)31-s + ⋯

Functional equation

Λ(s)=(432s/2 ΓC(s) L(s)=((−0.953−0.299i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(432s/2 ΓC(s+1/2) L(s)=((−0.953−0.299i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 432432    =    24⋅332^{4} \cdot 3^{3}
Sign: −0.953−0.299i-0.953 - 0.299i
Analytic conductor: 3.449533.44953
Root analytic conductor: 1.857291.85729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ432(97,⋅)\chi_{432} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 432, ( :1/2), −0.953−0.299i)(2,\ 432,\ (\ :1/2),\ -0.953 - 0.299i)

Particular Values

L(1)L(1) ≈\approx 0.0550685+0.358809i0.0550685 + 0.358809i
L(12)L(\frac12) ≈\approx 0.0550685+0.358809i0.0550685 + 0.358809i
L(32)L(\frac{3}{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+(0.159+1.72i)T 1 + (0.159 + 1.72i)T
good5 1+(2.08+0.757i)T+(3.83+3.21i)T2 1 + (2.08 + 0.757i)T + (3.83 + 3.21i)T^{2}
7 1+(−0.229+1.29i)T+(−6.57−2.39i)T2 1 + (-0.229 + 1.29i)T + (-6.57 - 2.39i)T^{2}
11 1+(4.90−1.78i)T+(8.42−7.07i)T2 1 + (4.90 - 1.78i)T + (8.42 - 7.07i)T^{2}
13 1+(0.0138−0.0116i)T+(2.25−12.8i)T2 1 + (0.0138 - 0.0116i)T + (2.25 - 12.8i)T^{2}
17 1+(−1.56−2.71i)T+(−8.5+14.7i)T2 1 + (-1.56 - 2.71i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.208+0.361i)T+(−9.5−16.4i)T2 1 + (-0.208 + 0.361i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.179−1.01i)T+(−21.6+7.86i)T2 1 + (-0.179 - 1.01i)T + (-21.6 + 7.86i)T^{2}
29 1+(5.98+5.01i)T+(5.03+28.5i)T2 1 + (5.98 + 5.01i)T + (5.03 + 28.5i)T^{2}
31 1+(0.647+3.67i)T+(−29.1+10.6i)T2 1 + (0.647 + 3.67i)T + (-29.1 + 10.6i)T^{2}
37 1+(2.21+3.83i)T+(−18.5+32.0i)T2 1 + (2.21 + 3.83i)T + (-18.5 + 32.0i)T^{2}
41 1+(2.81−2.36i)T+(7.11−40.3i)T2 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2}
43 1+(7.80−2.84i)T+(32.9−27.6i)T2 1 + (7.80 - 2.84i)T + (32.9 - 27.6i)T^{2}
47 1+(−1.23+6.99i)T+(−44.1−16.0i)T2 1 + (-1.23 + 6.99i)T + (-44.1 - 16.0i)T^{2}
53 1+1.30T+53T2 1 + 1.30T + 53T^{2}
59 1+(3.47+1.26i)T+(45.1+37.9i)T2 1 + (3.47 + 1.26i)T + (45.1 + 37.9i)T^{2}
61 1+(−1.20+6.80i)T+(−57.3−20.8i)T2 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2}
67 1+(−8.44+7.08i)T+(11.6−65.9i)T2 1 + (-8.44 + 7.08i)T + (11.6 - 65.9i)T^{2}
71 1+(3.04+5.26i)T+(−35.5+61.4i)T2 1 + (3.04 + 5.26i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.273+0.473i)T+(−36.5−63.2i)T2 1 + (-0.273 + 0.473i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.374+0.314i)T+(13.7+77.7i)T2 1 + (0.374 + 0.314i)T + (13.7 + 77.7i)T^{2}
83 1+(3.53+2.96i)T+(14.4+81.7i)T2 1 + (3.53 + 2.96i)T + (14.4 + 81.7i)T^{2}
89 1+(−1.68+2.92i)T+(−44.5−77.0i)T2 1 + (-1.68 + 2.92i)T + (-44.5 - 77.0i)T^{2}
97 1+(9.34−3.40i)T+(74.3−62.3i)T2 1 + (9.34 - 3.40i)T + (74.3 - 62.3i)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

−10.86332611348467389502656547410, −9.841767792706318413059663242773, −8.422798293255697037559667101990, −7.78823114872958657241899212462, −7.26116435525789464882263190615, −5.93552776429434520855729828633, −4.87336002057437841241032897766, −3.56948438096814540745352514423, −2.05009275352463876703087103378, −0.22001651112520579866512229005, 2.80628336165834024115115930579, 3.64530158848651561155314154203, 4.99545903069390012148633009199, 5.63467786620694440025394759881, 7.16103969980936713234807395979, 8.136129441955964089777237596589, 8.879303832397443560390604079161, 10.01595926747855761793539762011, 10.77931619700742347215009647866, 11.45384444035107643858698463408

Graph of the ZZ-function along the critical line