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Properties

Label 2-88-11.4-c3-0-8
Degree 22
Conductor 8888
Sign −0.993+0.109i-0.993 + 0.109i
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  + (2.02 − 6.21i)3-s + (−17.3 + 12.6i)5-s + (−9.35 − 28.7i)7-s + (−12.7 − 9.26i)9-s + (−27.5 + 23.9i)11-s + (−27.2 − 19.7i)13-s + (43.3 + 133. i)15-s + (−3.40 + 2.47i)17-s + (15.5 − 47.7i)19-s − 197.·21-s − 27.2·23-s + (103. − 318. i)25-s + (59.4 − 43.2i)27-s + (−7.51 − 23.1i)29-s + (−132. − 96.0i)31-s + ⋯
L(s)  = 1  + (0.388 − 1.19i)3-s + (−1.55 + 1.12i)5-s + (−0.504 − 1.55i)7-s + (−0.472 − 0.343i)9-s + (−0.755 + 0.655i)11-s + (−0.580 − 0.422i)13-s + (0.746 + 2.29i)15-s + (−0.0485 + 0.0352i)17-s + (0.187 − 0.576i)19-s − 2.05·21-s − 0.246·23-s + (0.828 − 2.54i)25-s + (0.423 − 0.307i)27-s + (−0.0481 − 0.148i)29-s + (−0.765 − 0.556i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((−0.993+0.109i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=((−0.993+0.109i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: −0.993+0.109i-0.993 + 0.109i
Analytic conductor: 5.192165.19216
Root analytic conductor: 2.278632.27863
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ88(81,⋅)\chi_{88} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :3/2), −0.993+0.109i)(2,\ 88,\ (\ :3/2),\ -0.993 + 0.109i)

Particular Values

L(2)L(2) ≈\approx 0.0306072−0.555824i0.0306072 - 0.555824i
L(12)L(\frac12) ≈\approx 0.0306072−0.555824i0.0306072 - 0.555824i
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+(27.5−23.9i)T 1 + (27.5 - 23.9i)T
good3 1+(−2.02+6.21i)T+(−21.8−15.8i)T2 1 + (-2.02 + 6.21i)T + (-21.8 - 15.8i)T^{2}
5 1+(17.3−12.6i)T+(38.6−118.i)T2 1 + (17.3 - 12.6i)T + (38.6 - 118. i)T^{2}
7 1+(9.35+28.7i)T+(−277.+201.i)T2 1 + (9.35 + 28.7i)T + (-277. + 201. i)T^{2}
13 1+(27.2+19.7i)T+(678.+2.08e3i)T2 1 + (27.2 + 19.7i)T + (678. + 2.08e3i)T^{2}
17 1+(3.40−2.47i)T+(1.51e3−4.67e3i)T2 1 + (3.40 - 2.47i)T + (1.51e3 - 4.67e3i)T^{2}
19 1+(−15.5+47.7i)T+(−5.54e3−4.03e3i)T2 1 + (-15.5 + 47.7i)T + (-5.54e3 - 4.03e3i)T^{2}
23 1+27.2T+1.21e4T2 1 + 27.2T + 1.21e4T^{2}
29 1+(7.51+23.1i)T+(−1.97e4+1.43e4i)T2 1 + (7.51 + 23.1i)T + (-1.97e4 + 1.43e4i)T^{2}
31 1+(132.+96.0i)T+(9.20e3+2.83e4i)T2 1 + (132. + 96.0i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(−96.0−295.i)T+(−4.09e4+2.97e4i)T2 1 + (-96.0 - 295. i)T + (-4.09e4 + 2.97e4i)T^{2}
41 1+(−124.+381.i)T+(−5.57e4−4.05e4i)T2 1 + (-124. + 381. i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1−216.T+7.95e4T2 1 - 216.T + 7.95e4T^{2}
47 1+(−31.5+96.9i)T+(−8.39e4−6.10e4i)T2 1 + (-31.5 + 96.9i)T + (-8.39e4 - 6.10e4i)T^{2}
53 1+(190.+138.i)T+(4.60e4+1.41e5i)T2 1 + (190. + 138. i)T + (4.60e4 + 1.41e5i)T^{2}
59 1+(39.0+120.i)T+(−1.66e5+1.20e5i)T2 1 + (39.0 + 120. i)T + (-1.66e5 + 1.20e5i)T^{2}
61 1+(206.−150.i)T+(7.01e4−2.15e5i)T2 1 + (206. - 150. i)T + (7.01e4 - 2.15e5i)T^{2}
67 1+332.T+3.00e5T2 1 + 332.T + 3.00e5T^{2}
71 1+(273.−198.i)T+(1.10e5−3.40e5i)T2 1 + (273. - 198. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(−166.−512.i)T+(−3.14e5+2.28e5i)T2 1 + (-166. - 512. i)T + (-3.14e5 + 2.28e5i)T^{2}
79 1+(−38.7−28.1i)T+(1.52e5+4.68e5i)T2 1 + (-38.7 - 28.1i)T + (1.52e5 + 4.68e5i)T^{2}
83 1+(−19.7+14.3i)T+(1.76e5−5.43e5i)T2 1 + (-19.7 + 14.3i)T + (1.76e5 - 5.43e5i)T^{2}
89 1+1.49e3T+7.04e5T2 1 + 1.49e3T + 7.04e5T^{2}
97 1+(−724.−526.i)T+(2.82e5+8.68e5i)T2 1 + (-724. - 526. 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

−13.14921860885921560503736898468, −12.24726082179660229180611590371, −11.00758719698865483595372744611, −10.13611919997651020238545049243, −7.939370221290429074520758203845, −7.38531887833777044535158336603, −6.83801611618513231828769464410, −4.18251840142318773081489374316, −2.81202363370593926608147125136, −0.30896169002472842178713129213, 3.15656343576287382206357898660, 4.42826013215091722393491553447, 5.52748367696184595338870571464, 7.81973054686149543531310118947, 8.836439510366847845561638250795, 9.423253912365042449869323270587, 11.02389629483686473429552488659, 12.13633650336726353284790799191, 12.76559837960113954309572929550, 14.63500254627137093636591707977

Graph of the ZZ-function along the critical line