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Properties

Label 2-88-11.5-c3-0-4
Degree 22
Conductor 8888
Sign 0.389+0.921i0.389 + 0.921i
Analytic cond. 5.192165.19216
Root an. cond. 2.278632.27863
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  + (−2.34 − 1.70i)3-s + (−2.01 + 6.21i)5-s + (20.0 − 14.5i)7-s + (−5.74 − 17.6i)9-s + (32.9 − 15.7i)11-s + (−22.3 − 68.7i)13-s + (15.3 − 11.1i)15-s + (−1.71 + 5.28i)17-s + (−31.7 − 23.0i)19-s − 71.9·21-s + 172.·23-s + (66.5 + 48.3i)25-s + (−40.8 + 125. i)27-s + (−11.0 + 7.99i)29-s + (−32.5 − 100. i)31-s + ⋯
L(s)  = 1  + (−0.451 − 0.327i)3-s + (−0.180 + 0.555i)5-s + (1.08 − 0.787i)7-s + (−0.212 − 0.655i)9-s + (0.902 − 0.431i)11-s + (−0.476 − 1.46i)13-s + (0.263 − 0.191i)15-s + (−0.0244 + 0.0753i)17-s + (−0.383 − 0.278i)19-s − 0.747·21-s + 1.55·23-s + (0.532 + 0.387i)25-s + (−0.291 + 0.895i)27-s + (−0.0704 + 0.0511i)29-s + (−0.188 − 0.579i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((0.389+0.921i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=((0.389+0.921i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 0.389+0.921i0.389 + 0.921i
Analytic conductor: 5.192165.19216
Root analytic conductor: 2.278632.27863
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ88(49,⋅)\chi_{88} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :3/2), 0.389+0.921i)(2,\ 88,\ (\ :3/2),\ 0.389 + 0.921i)

Particular Values

L(2)L(2) ≈\approx 1.10844−0.734855i1.10844 - 0.734855i
L(12)L(\frac12) ≈\approx 1.10844−0.734855i1.10844 - 0.734855i
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
11 1+(−32.9+15.7i)T 1 + (-32.9 + 15.7i)T
good3 1+(2.34+1.70i)T+(8.34+25.6i)T2 1 + (2.34 + 1.70i)T + (8.34 + 25.6i)T^{2}
5 1+(2.01−6.21i)T+(−101.−73.4i)T2 1 + (2.01 - 6.21i)T + (-101. - 73.4i)T^{2}
7 1+(−20.0+14.5i)T+(105.−326.i)T2 1 + (-20.0 + 14.5i)T + (105. - 326. i)T^{2}
13 1+(22.3+68.7i)T+(−1.77e3+1.29e3i)T2 1 + (22.3 + 68.7i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(1.71−5.28i)T+(−3.97e3−2.88e3i)T2 1 + (1.71 - 5.28i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(31.7+23.0i)T+(2.11e3+6.52e3i)T2 1 + (31.7 + 23.0i)T + (2.11e3 + 6.52e3i)T^{2}
23 1−172.T+1.21e4T2 1 - 172.T + 1.21e4T^{2}
29 1+(11.0−7.99i)T+(7.53e3−2.31e4i)T2 1 + (11.0 - 7.99i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(32.5+100.i)T+(−2.41e4+1.75e4i)T2 1 + (32.5 + 100. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(352.−255.i)T+(1.56e4−4.81e4i)T2 1 + (352. - 255. i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(53.1+38.6i)T+(2.12e4+6.55e4i)T2 1 + (53.1 + 38.6i)T + (2.12e4 + 6.55e4i)T^{2}
43 1−230.T+7.95e4T2 1 - 230.T + 7.95e4T^{2}
47 1+(198.+144.i)T+(3.20e4+9.87e4i)T2 1 + (198. + 144. i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(−92.7−285.i)T+(−1.20e5+8.75e4i)T2 1 + (-92.7 - 285. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(−71.1+51.7i)T+(6.34e4−1.95e5i)T2 1 + (-71.1 + 51.7i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(103.−317.i)T+(−1.83e5−1.33e5i)T2 1 + (103. - 317. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1+345.T+3.00e5T2 1 + 345.T + 3.00e5T^{2}
71 1+(−203.+624.i)T+(−2.89e5−2.10e5i)T2 1 + (-203. + 624. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−860.+624.i)T+(1.20e5−3.69e5i)T2 1 + (-860. + 624. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(−165.−509.i)T+(−3.98e5+2.89e5i)T2 1 + (-165. - 509. i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(9.74−29.9i)T+(−4.62e5−3.36e5i)T2 1 + (9.74 - 29.9i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1−1.20e3T+7.04e5T2 1 - 1.20e3T + 7.04e5T^{2}
97 1+(161.+497.i)T+(−7.38e5+5.36e5i)T2 1 + (161. + 497. i)T + (-7.38e5 + 5.36e5i)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

−13.48327619396251668733296095514, −12.27629993087970180732405638543, −11.22478418756824250995112076541, −10.57534767381386964986350783025, −8.919492571572108113081388676510, −7.58485766236519429234647669243, −6.59903691374715700684785638120, −5.10116811530944817512523194945, −3.38149642647001139748849088613, −0.932410038234645167240944775196, 1.88416715320367914411331595462, 4.46207636099811113800106520716, 5.24295221652677244978595091511, 6.93229181848011103758389616605, 8.472763766129043200161368335204, 9.274725401750217466290569464635, 10.88194119105149220789590992283, 11.69864774083889686454124050031, 12.50021731241442672317920657727, 14.11890824432242874803531652889

Graph of the ZZ-function along the critical line