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Properties

Label 2-110-11.3-c3-0-0
Degree 22
Conductor 110110
Sign −0.969+0.246i-0.969 + 0.246i
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  + (−1.61 + 1.17i)2-s + (2.75 + 8.48i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (−14.4 − 10.4i)6-s + (−4.06 + 12.5i)7-s + (2.47 + 7.60i)8-s + (−42.4 + 30.8i)9-s + 10·10-s + (−22.6 + 28.6i)11-s + 35.6·12-s + (30.1 − 21.9i)13-s + (−8.13 − 25.0i)14-s + (13.7 − 42.4i)15-s + (−12.9 − 9.40i)16-s + (−89.8 − 65.3i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.530 + 1.63i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.981 − 0.713i)6-s + (−0.219 + 0.676i)7-s + (0.109 + 0.336i)8-s + (−1.57 + 1.14i)9-s + 0.316·10-s + (−0.620 + 0.784i)11-s + 0.858·12-s + (0.643 − 0.467i)13-s + (−0.155 − 0.478i)14-s + (0.237 − 0.729i)15-s + (−0.202 − 0.146i)16-s + (−1.28 − 0.931i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.969+0.246i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.969+0.246i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.969+0.246i-0.969 + 0.246i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(91,⋅)\chi_{110} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), −0.969+0.246i)(2,\ 110,\ (\ :3/2),\ -0.969 + 0.246i)

Particular Values

L(2)L(2) ≈\approx 0.111396−0.889443i0.111396 - 0.889443i
L(12)L(\frac12) ≈\approx 0.111396−0.889443i0.111396 - 0.889443i
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+(1.61−1.17i)T 1 + (1.61 - 1.17i)T
5 1+(4.04+2.93i)T 1 + (4.04 + 2.93i)T
11 1+(22.6−28.6i)T 1 + (22.6 - 28.6i)T
good3 1+(−2.75−8.48i)T+(−21.8+15.8i)T2 1 + (-2.75 - 8.48i)T + (-21.8 + 15.8i)T^{2}
7 1+(4.06−12.5i)T+(−277.−201.i)T2 1 + (4.06 - 12.5i)T + (-277. - 201. i)T^{2}
13 1+(−30.1+21.9i)T+(678.−2.08e3i)T2 1 + (-30.1 + 21.9i)T + (678. - 2.08e3i)T^{2}
17 1+(89.8+65.3i)T+(1.51e3+4.67e3i)T2 1 + (89.8 + 65.3i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(−4.51−13.9i)T+(−5.54e3+4.03e3i)T2 1 + (-4.51 - 13.9i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−115.T+1.21e4T2 1 - 115.T + 1.21e4T^{2}
29 1+(27.4−84.3i)T+(−1.97e4−1.43e4i)T2 1 + (27.4 - 84.3i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(187.−136.i)T+(9.20e3−2.83e4i)T2 1 + (187. - 136. i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(35.0−107.i)T+(−4.09e4−2.97e4i)T2 1 + (35.0 - 107. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(76.4+235.i)T+(−5.57e4+4.05e4i)T2 1 + (76.4 + 235. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−515.T+7.95e4T2 1 - 515.T + 7.95e4T^{2}
47 1+(−140.−433.i)T+(−8.39e4+6.10e4i)T2 1 + (-140. - 433. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(423.−307.i)T+(4.60e4−1.41e5i)T2 1 + (423. - 307. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(205.−633.i)T+(−1.66e5−1.20e5i)T2 1 + (205. - 633. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−534.−388.i)T+(7.01e4+2.15e5i)T2 1 + (-534. - 388. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−250.T+3.00e5T2 1 - 250.T + 3.00e5T^{2}
71 1+(480.+348.i)T+(1.10e5+3.40e5i)T2 1 + (480. + 348. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(114.−351.i)T+(−3.14e5−2.28e5i)T2 1 + (114. - 351. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(42.8−31.1i)T+(1.52e5−4.68e5i)T2 1 + (42.8 - 31.1i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−97.6−70.9i)T+(1.76e5+5.43e5i)T2 1 + (-97.6 - 70.9i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+308.T+7.04e5T2 1 + 308.T + 7.04e5T^{2}
97 1+(−1.13e3+827.i)T+(2.82e5−8.68e5i)T2 1 + (-1.13e3 + 827. 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

−14.14858672599048319745728969917, −12.73754488324799772949984363602, −11.14943607240331795881868248875, −10.41793934802532273933545290692, −9.102542136718754068826686876244, −8.895655914603639466237322029325, −7.39608625150754877526191142022, −5.51112973487993451903397347953, −4.48569264321580585308730226090, −2.83698525724171550229723781892, 0.53945698517672758868091614437, 2.14656679710626155014866764338, 3.61592363421056632079031212342, 6.31550153523085924872067962589, 7.23303017381603742202126246468, 8.166510661083178972841997063705, 9.031537116189002221348299621544, 10.81391341813184069843814777204, 11.45956454155948261106852707268, 12.96853072122802934594090747551

Graph of the ZZ-function along the critical line