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Properties

Label 2-88-11.3-c3-0-1
Degree 22
Conductor 8888
Sign −0.0924−0.995i-0.0924 - 0.995i
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  + (1.71 + 5.26i)3-s + (5.02 + 3.65i)5-s + (−3.12 + 9.60i)7-s + (−2.98 + 2.17i)9-s + (−17.8 + 31.8i)11-s + (−16.0 + 11.6i)13-s + (−10.6 + 32.7i)15-s + (−0.0229 − 0.0166i)17-s + (22.3 + 68.8i)19-s − 55.9·21-s + 92.3·23-s + (−26.6 − 82.1i)25-s + (104. + 75.8i)27-s + (25.7 − 79.2i)29-s + (139. − 101. i)31-s + ⋯
L(s)  = 1  + (0.329 + 1.01i)3-s + (0.449 + 0.326i)5-s + (−0.168 + 0.518i)7-s + (−0.110 + 0.0804i)9-s + (−0.489 + 0.871i)11-s + (−0.342 + 0.248i)13-s + (−0.183 + 0.563i)15-s + (−0.000327 − 0.000237i)17-s + (0.270 + 0.831i)19-s − 0.581·21-s + 0.837·23-s + (−0.213 − 0.656i)25-s + (0.744 + 0.540i)27-s + (0.164 − 0.507i)29-s + (0.806 − 0.585i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((−0.0924−0.995i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0924 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=((−0.0924−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0924 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: −0.0924−0.995i-0.0924 - 0.995i
Analytic conductor: 5.192165.19216
Root analytic conductor: 2.278632.27863
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ88(25,⋅)\chi_{88} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :3/2), −0.0924−0.995i)(2,\ 88,\ (\ :3/2),\ -0.0924 - 0.995i)

Particular Values

L(2)L(2) ≈\approx 1.13840+1.24895i1.13840 + 1.24895i
L(12)L(\frac12) ≈\approx 1.13840+1.24895i1.13840 + 1.24895i
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+(17.8−31.8i)T 1 + (17.8 - 31.8i)T
good3 1+(−1.71−5.26i)T+(−21.8+15.8i)T2 1 + (-1.71 - 5.26i)T + (-21.8 + 15.8i)T^{2}
5 1+(−5.02−3.65i)T+(38.6+118.i)T2 1 + (-5.02 - 3.65i)T + (38.6 + 118. i)T^{2}
7 1+(3.12−9.60i)T+(−277.−201.i)T2 1 + (3.12 - 9.60i)T + (-277. - 201. i)T^{2}
13 1+(16.0−11.6i)T+(678.−2.08e3i)T2 1 + (16.0 - 11.6i)T + (678. - 2.08e3i)T^{2}
17 1+(0.0229+0.0166i)T+(1.51e3+4.67e3i)T2 1 + (0.0229 + 0.0166i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(−22.3−68.8i)T+(−5.54e3+4.03e3i)T2 1 + (-22.3 - 68.8i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−92.3T+1.21e4T2 1 - 92.3T + 1.21e4T^{2}
29 1+(−25.7+79.2i)T+(−1.97e4−1.43e4i)T2 1 + (-25.7 + 79.2i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−139.+101.i)T+(9.20e3−2.83e4i)T2 1 + (-139. + 101. i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−55.1+169.i)T+(−4.09e4−2.97e4i)T2 1 + (-55.1 + 169. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(72.8+224.i)T+(−5.57e4+4.05e4i)T2 1 + (72.8 + 224. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+69.3T+7.95e4T2 1 + 69.3T + 7.95e4T^{2}
47 1+(133.+411.i)T+(−8.39e4+6.10e4i)T2 1 + (133. + 411. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−485.+352.i)T+(4.60e4−1.41e5i)T2 1 + (-485. + 352. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(112.−347.i)T+(−1.66e5−1.20e5i)T2 1 + (112. - 347. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−554.−402.i)T+(7.01e4+2.15e5i)T2 1 + (-554. - 402. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−224.T+3.00e5T2 1 - 224.T + 3.00e5T^{2}
71 1+(−206.−149.i)T+(1.10e5+3.40e5i)T2 1 + (-206. - 149. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(272.−839.i)T+(−3.14e5−2.28e5i)T2 1 + (272. - 839. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(313.−227.i)T+(1.52e5−4.68e5i)T2 1 + (313. - 227. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(982.+713.i)T+(1.76e5+5.43e5i)T2 1 + (982. + 713. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+1.23e3T+7.04e5T2 1 + 1.23e3T + 7.04e5T^{2}
97 1+(−721.+524.i)T+(2.82e5−8.68e5i)T2 1 + (-721. + 524. i)T + (2.82e5 - 8.68e5i)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

−14.21285530035069622419042089344, −12.90766741748512983894250191087, −11.79118005145428862722249363959, −10.23203598063402823320284871510, −9.831853029299234126893791894975, −8.622217832936667002372562717030, −7.04911331594021165373326494413, −5.50119729924340288839096405579, −4.14217610226560311000014978732, −2.49434854920318690313435961363, 1.07287976651627275233622879383, 2.89244284341014989178280446935, 5.04355881376770893518048665774, 6.57742383281998157294935975027, 7.62846935234673218226195952226, 8.734000865637953971500064971707, 10.07611043391351073644625214451, 11.30141720585778367886427184685, 12.70584201183665267505144359039, 13.35824624598775136862036297993

Graph of the ZZ-function along the critical line