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Properties

Label 2-110-11.9-c3-0-1
Degree 22
Conductor 110110
Sign 0.371−0.928i0.371 - 0.928i
Analytic cond. 6.490216.49021
Root an. cond. 2.547582.54758
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  + (0.618 − 1.90i)2-s + (−3.92 + 2.85i)3-s + (−3.23 − 2.35i)4-s + (−1.54 − 4.75i)5-s + (3.00 + 9.23i)6-s + (5.32 + 3.86i)7-s + (−6.47 + 4.70i)8-s + (−1.05 + 3.25i)9-s − 10.0·10-s + (18.0 + 31.7i)11-s + 19.4·12-s + (−18.7 + 57.6i)13-s + (10.6 − 7.73i)14-s + (19.6 + 14.2i)15-s + (4.94 + 15.2i)16-s + (26.5 + 81.8i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.755 + 0.549i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.204 + 0.628i)6-s + (0.287 + 0.208i)7-s + (−0.286 + 0.207i)8-s + (−0.0391 + 0.120i)9-s − 0.316·10-s + (0.493 + 0.869i)11-s + 0.467·12-s + (−0.399 + 1.22i)13-s + (0.203 − 0.147i)14-s + (0.338 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.379 + 1.16i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.371−0.928i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((0.371−0.928i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.371−0.928i0.371 - 0.928i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(31,⋅)\chi_{110} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), 0.371−0.928i)(2,\ 110,\ (\ :3/2),\ 0.371 - 0.928i)

Particular Values

L(2)L(2) ≈\approx 0.794833+0.538071i0.794833 + 0.538071i
L(12)L(\frac12) ≈\approx 0.794833+0.538071i0.794833 + 0.538071i
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+(−0.618+1.90i)T 1 + (-0.618 + 1.90i)T
5 1+(1.54+4.75i)T 1 + (1.54 + 4.75i)T
11 1+(−18.0−31.7i)T 1 + (-18.0 - 31.7i)T
good3 1+(3.92−2.85i)T+(8.34−25.6i)T2 1 + (3.92 - 2.85i)T + (8.34 - 25.6i)T^{2}
7 1+(−5.32−3.86i)T+(105.+326.i)T2 1 + (-5.32 - 3.86i)T + (105. + 326. i)T^{2}
13 1+(18.7−57.6i)T+(−1.77e3−1.29e3i)T2 1 + (18.7 - 57.6i)T + (-1.77e3 - 1.29e3i)T^{2}
17 1+(−26.5−81.8i)T+(−3.97e3+2.88e3i)T2 1 + (-26.5 - 81.8i)T + (-3.97e3 + 2.88e3i)T^{2}
19 1+(−2.21+1.60i)T+(2.11e3−6.52e3i)T2 1 + (-2.21 + 1.60i)T + (2.11e3 - 6.52e3i)T^{2}
23 1+70.7T+1.21e4T2 1 + 70.7T + 1.21e4T^{2}
29 1+(2.67+1.94i)T+(7.53e3+2.31e4i)T2 1 + (2.67 + 1.94i)T + (7.53e3 + 2.31e4i)T^{2}
31 1+(−30.9+95.2i)T+(−2.41e4−1.75e4i)T2 1 + (-30.9 + 95.2i)T + (-2.41e4 - 1.75e4i)T^{2}
37 1+(−264.−191.i)T+(1.56e4+4.81e4i)T2 1 + (-264. - 191. i)T + (1.56e4 + 4.81e4i)T^{2}
41 1+(16.6−12.1i)T+(2.12e4−6.55e4i)T2 1 + (16.6 - 12.1i)T + (2.12e4 - 6.55e4i)T^{2}
43 1+469.T+7.95e4T2 1 + 469.T + 7.95e4T^{2}
47 1+(238.−173.i)T+(3.20e4−9.87e4i)T2 1 + (238. - 173. i)T + (3.20e4 - 9.87e4i)T^{2}
53 1+(121.−373.i)T+(−1.20e5−8.75e4i)T2 1 + (121. - 373. i)T + (-1.20e5 - 8.75e4i)T^{2}
59 1+(−477.−347.i)T+(6.34e4+1.95e5i)T2 1 + (-477. - 347. i)T + (6.34e4 + 1.95e5i)T^{2}
61 1+(70.0+215.i)T+(−1.83e5+1.33e5i)T2 1 + (70.0 + 215. i)T + (-1.83e5 + 1.33e5i)T^{2}
67 1+177.T+3.00e5T2 1 + 177.T + 3.00e5T^{2}
71 1+(163.+504.i)T+(−2.89e5+2.10e5i)T2 1 + (163. + 504. i)T + (-2.89e5 + 2.10e5i)T^{2}
73 1+(−734.−533.i)T+(1.20e5+3.69e5i)T2 1 + (-734. - 533. i)T + (1.20e5 + 3.69e5i)T^{2}
79 1+(−260.+802.i)T+(−3.98e5−2.89e5i)T2 1 + (-260. + 802. i)T + (-3.98e5 - 2.89e5i)T^{2}
83 1+(−149.−460.i)T+(−4.62e5+3.36e5i)T2 1 + (-149. - 460. i)T + (-4.62e5 + 3.36e5i)T^{2}
89 1+60.8T+7.04e5T2 1 + 60.8T + 7.04e5T^{2}
97 1+(−79.2+244.i)T+(−7.38e5−5.36e5i)T2 1 + (-79.2 + 244. 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

−13.15134169803468967873329205906, −11.97194568006056436837335815168, −11.54603408754412595480618450857, −10.28252765980396979104157359714, −9.465403520533077540722165733843, −8.094629874497301186882541504435, −6.30155745778655762512825432089, −4.93545913443420567199605617826, −4.13016955139014695709850161162, −1.83811215170196671675891062311, 0.54949703419263031430273600822, 3.33692212565432408276811844161, 5.18690494789197914724195961667, 6.20268149792233739495497777644, 7.22286596416538320112581942853, 8.265416626337633058295149022305, 9.778579625802917681215842340039, 11.16906217788216769574632096699, 11.96107013938567183466456297510, 13.01693109184223849865042057452

Graph of the ZZ-function along the critical line