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Properties

Label 2-110-11.3-c3-0-7
Degree 22
Conductor 110110
Sign −0.193+0.981i-0.193 + 0.981i
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 + (−1.38 − 4.24i)3-s + (1.23 − 3.80i)4-s + (4.04 + 2.93i)5-s + (7.22 + 5.25i)6-s + (−0.607 + 1.86i)7-s + (2.47 + 7.60i)8-s + (5.69 − 4.13i)9-s − 10·10-s + (−16.0 − 32.7i)11-s − 17.8·12-s + (−10.8 + 7.88i)13-s + (−1.21 − 3.73i)14-s + (6.90 − 21.2i)15-s + (−12.9 − 9.40i)16-s + (−93.2 − 67.7i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.265 − 0.817i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.491 + 0.357i)6-s + (−0.0327 + 0.100i)7-s + (0.109 + 0.336i)8-s + (0.210 − 0.153i)9-s − 0.316·10-s + (−0.441 − 0.897i)11-s − 0.429·12-s + (−0.231 + 0.168i)13-s + (−0.0231 − 0.0713i)14-s + (0.118 − 0.365i)15-s + (−0.202 − 0.146i)16-s + (−1.33 − 0.966i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.193+0.981i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.193+0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.193+0.981i-0.193 + 0.981i
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), −0.193+0.981i)(2,\ 110,\ (\ :3/2),\ -0.193 + 0.981i)

Particular Values

L(2)L(2) ≈\approx 0.549458−0.668743i0.549458 - 0.668743i
L(12)L(\frac12) ≈\approx 0.549458−0.668743i0.549458 - 0.668743i
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+(16.0+32.7i)T 1 + (16.0 + 32.7i)T
good3 1+(1.38+4.24i)T+(−21.8+15.8i)T2 1 + (1.38 + 4.24i)T + (-21.8 + 15.8i)T^{2}
7 1+(0.607−1.86i)T+(−277.−201.i)T2 1 + (0.607 - 1.86i)T + (-277. - 201. i)T^{2}
13 1+(10.8−7.88i)T+(678.−2.08e3i)T2 1 + (10.8 - 7.88i)T + (678. - 2.08e3i)T^{2}
17 1+(93.2+67.7i)T+(1.51e3+4.67e3i)T2 1 + (93.2 + 67.7i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(44.0+135.i)T+(−5.54e3+4.03e3i)T2 1 + (44.0 + 135. i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−158.T+1.21e4T2 1 - 158.T + 1.21e4T^{2}
29 1+(−39.9+122.i)T+(−1.97e4−1.43e4i)T2 1 + (-39.9 + 122. i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−11.0+7.99i)T+(9.20e3−2.83e4i)T2 1 + (-11.0 + 7.99i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(118.−365.i)T+(−4.09e4−2.97e4i)T2 1 + (118. - 365. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(55.9+172.i)T+(−5.57e4+4.05e4i)T2 1 + (55.9 + 172. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+199.T+7.95e4T2 1 + 199.T + 7.95e4T^{2}
47 1+(30.9+95.2i)T+(−8.39e4+6.10e4i)T2 1 + (30.9 + 95.2i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(130.−94.4i)T+(4.60e4−1.41e5i)T2 1 + (130. - 94.4i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−144.+444.i)T+(−1.66e5−1.20e5i)T2 1 + (-144. + 444. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−579.−420.i)T+(7.01e4+2.15e5i)T2 1 + (-579. - 420. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−844.T+3.00e5T2 1 - 844.T + 3.00e5T^{2}
71 1+(−168.−122.i)T+(1.10e5+3.40e5i)T2 1 + (-168. - 122. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(51.0−157.i)T+(−3.14e5−2.28e5i)T2 1 + (51.0 - 157. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−933.+678.i)T+(1.52e5−4.68e5i)T2 1 + (-933. + 678. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(233.+169.i)T+(1.76e5+5.43e5i)T2 1 + (233. + 169. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−181.T+7.04e5T2 1 - 181.T + 7.04e5T^{2}
97 1+(−371.+269.i)T+(2.82e5−8.68e5i)T2 1 + (-371. + 269. 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

−13.20454430512003904572062796918, −11.66343838159726760163804691365, −10.85513533215892925725897759744, −9.480332307336773305930941961711, −8.511943109108459532313536483283, −7.02864715151737907019530649746, −6.54712741548742782712809445395, −5.01490472347821962335836297132, −2.50533952858152833386837121662, −0.57986836463863395141316239489, 1.93452712643048490458752034847, 3.97674985518514192265760764559, 5.18979198249897867848906790337, 6.89348819102457583481031484948, 8.298563470958815006595567557311, 9.450216862940904127594602840275, 10.35703366038101939533635151140, 10.92386948301547131530902114606, 12.49281956019886379658156678095, 13.09329742092234896384964761634

Graph of the ZZ-function along the critical line