↑

Properties

Label 2-54-9.2-c2-0-0
Degree 22
Conductor 5454
Sign 0.703+0.710i0.703 + 0.710i
Analytic cond. 1.471391.47139
Root an. cond. 1.213011.21301
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (4.5 − 2.59i)5-s + (4.17 − 7.22i)7-s − 2.82i·8-s − 7.34·10-s + (−0.825 − 0.476i)11-s + (4.84 + 8.39i)13-s + (−10.2 + 5.90i)14-s + (−2.00 + 3.46i)16-s + 18.8i·17-s − 24.6·19-s + (8.99 + 5.19i)20-s + (0.674 + 1.16i)22-s + (−0.825 + 0.476i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.900 − 0.519i)5-s + (0.596 − 1.03i)7-s − 0.353i·8-s − 0.734·10-s + (−0.0750 − 0.0433i)11-s + (0.372 + 0.645i)13-s + (−0.730 + 0.421i)14-s + (−0.125 + 0.216i)16-s + 1.11i·17-s − 1.29·19-s + (0.449 + 0.259i)20-s + (0.0306 + 0.0530i)22-s + (−0.0359 + 0.0207i)23-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((0.703+0.710i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((0.703+0.710i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: 0.703+0.710i0.703 + 0.710i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(35,⋅)\chi_{54} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), 0.703+0.710i)(2,\ 54,\ (\ :1),\ 0.703 + 0.710i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.935931−0.390207i0.935931 - 0.390207i
L(12)L(\frac12) ≈\approx 0.935931−0.390207i0.935931 - 0.390207i
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+(1.22+0.707i)T 1 + (1.22 + 0.707i)T
3 1 1
good5 1+(−4.5+2.59i)T+(12.5−21.6i)T2 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2}
7 1+(−4.17+7.22i)T+(−24.5−42.4i)T2 1 + (-4.17 + 7.22i)T + (-24.5 - 42.4i)T^{2}
11 1+(0.825+0.476i)T+(60.5+104.i)T2 1 + (0.825 + 0.476i)T + (60.5 + 104. i)T^{2}
13 1+(−4.84−8.39i)T+(−84.5+146.i)T2 1 + (-4.84 - 8.39i)T + (-84.5 + 146. i)T^{2}
17 1−18.8iT−289T2 1 - 18.8iT - 289T^{2}
19 1+24.6T+361T2 1 + 24.6T + 361T^{2}
23 1+(0.825−0.476i)T+(264.5−458.i)T2 1 + (0.825 - 0.476i)T + (264.5 - 458. i)T^{2}
29 1+(11.8+6.84i)T+(420.5+728.i)T2 1 + (11.8 + 6.84i)T + (420.5 + 728. i)T^{2}
31 1+(1.52+2.63i)T+(−480.5+832.i)T2 1 + (1.52 + 2.63i)T + (-480.5 + 832. i)T^{2}
37 1−46.6T+1.36e3T2 1 - 46.6T + 1.36e3T^{2}
41 1+(−9.45+5.45i)T+(840.5−1.45e3i)T2 1 + (-9.45 + 5.45i)T + (840.5 - 1.45e3i)T^{2}
43 1+(22.5−39.0i)T+(−924.5−1.60e3i)T2 1 + (22.5 - 39.0i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(39.2+22.6i)T+(1.10e3+1.91e3i)T2 1 + (39.2 + 22.6i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−94.3iT−2.80e3T2 1 - 94.3iT - 2.80e3T^{2}
59 1+(−16.2+9.39i)T+(1.74e3−3.01e3i)T2 1 + (-16.2 + 9.39i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(6.54−11.3i)T+(−1.86e3−3.22e3i)T2 1 + (6.54 - 11.3i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(37.5+64.9i)T+(−2.24e3+3.88e3i)T2 1 + (37.5 + 64.9i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+18.0iT−5.04e3T2 1 + 18.0iT - 5.04e3T^{2}
73 1+7.90T+5.32e3T2 1 + 7.90T + 5.32e3T^{2}
79 1+(−21.8+37.8i)T+(−3.12e3−5.40e3i)T2 1 + (-21.8 + 37.8i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−112.−65.1i)T+(3.44e3+5.96e3i)T2 1 + (-112. - 65.1i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+145.iT−7.92e3T2 1 + 145. iT - 7.92e3T^{2}
97 1+(−54.9+95.1i)T+(−4.70e3−8.14e3i)T2 1 + (-54.9 + 95.1i)T + (-4.70e3 - 8.14e3i)T^{2}
show more
show less
   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

−14.91651964856648928465677083397, −13.65281874160969723183124765667, −12.79672110576093955537705909403, −11.21624057442467680263613425235, −10.30932528085668470810545272604, −9.093540035510732274424143048834, −7.896440313459928808872460266923, −6.25659367932188497082465774961, −4.27572217217527577589462190887, −1.65153521327229550390934101392, 2.33306185666327441288755425339, 5.33892658796957644741213937444, 6.49664812410755589559403732794, 8.124185686916382445432521365774, 9.268836324445037925403635260131, 10.43669499780677752900430125500, 11.58493321099851535231019597427, 13.11915203262088770438518397429, 14.44690765156459786867907858091, 15.19424444203291778637267969725

Graph of the ZZ-function along the critical line