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Properties

Label 2-110-11.3-c3-0-8
Degree 22
Conductor 110110
Sign −0.0120+0.999i-0.0120 + 0.999i
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 + (0.168 + 0.519i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (0.884 + 0.642i)6-s + (3.58 − 11.0i)7-s + (−2.47 − 7.60i)8-s + (21.6 − 15.6i)9-s − 10·10-s + (−21.7 − 29.2i)11-s + 2.18·12-s + (29.3 − 21.2i)13-s + (−7.17 − 22.0i)14-s + (0.844 − 2.59i)15-s + (−12.9 − 9.40i)16-s + (−19.2 − 13.9i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0324 + 0.100i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.0601 + 0.0437i)6-s + (0.193 − 0.596i)7-s + (−0.109 − 0.336i)8-s + (0.800 − 0.581i)9-s − 0.316·10-s + (−0.597 − 0.802i)11-s + 0.0525·12-s + (0.625 − 0.454i)13-s + (−0.136 − 0.421i)14-s + (0.0145 − 0.0447i)15-s + (−0.202 − 0.146i)16-s + (−0.274 − 0.199i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.0120+0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.0120+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.0120+0.999i-0.0120 + 0.999i
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), −0.0120+0.999i)(2,\ 110,\ (\ :3/2),\ -0.0120 + 0.999i)

Particular Values

L(2)L(2) ≈\approx 1.43994−1.45739i1.43994 - 1.45739i
L(12)L(\frac12) ≈\approx 1.43994−1.45739i1.43994 - 1.45739i
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+(21.7+29.2i)T 1 + (21.7 + 29.2i)T
good3 1+(−0.168−0.519i)T+(−21.8+15.8i)T2 1 + (-0.168 - 0.519i)T + (-21.8 + 15.8i)T^{2}
7 1+(−3.58+11.0i)T+(−277.−201.i)T2 1 + (-3.58 + 11.0i)T + (-277. - 201. i)T^{2}
13 1+(−29.3+21.2i)T+(678.−2.08e3i)T2 1 + (-29.3 + 21.2i)T + (678. - 2.08e3i)T^{2}
17 1+(19.2+13.9i)T+(1.51e3+4.67e3i)T2 1 + (19.2 + 13.9i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(−20.4−63.0i)T+(−5.54e3+4.03e3i)T2 1 + (-20.4 - 63.0i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−23.2T+1.21e4T2 1 - 23.2T + 1.21e4T^{2}
29 1+(16.4−50.5i)T+(−1.97e4−1.43e4i)T2 1 + (16.4 - 50.5i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−32.3+23.5i)T+(9.20e3−2.83e4i)T2 1 + (-32.3 + 23.5i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(72.7−223.i)T+(−4.09e4−2.97e4i)T2 1 + (72.7 - 223. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−58.7−180.i)T+(−5.57e4+4.05e4i)T2 1 + (-58.7 - 180. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−113.T+7.95e4T2 1 - 113.T + 7.95e4T^{2}
47 1+(−22.5−69.4i)T+(−8.39e4+6.10e4i)T2 1 + (-22.5 - 69.4i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(−397.+288.i)T+(4.60e4−1.41e5i)T2 1 + (-397. + 288. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(187.−575.i)T+(−1.66e5−1.20e5i)T2 1 + (187. - 575. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−584.−424.i)T+(7.01e4+2.15e5i)T2 1 + (-584. - 424. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1+380.T+3.00e5T2 1 + 380.T + 3.00e5T^{2}
71 1+(350.+254.i)T+(1.10e5+3.40e5i)T2 1 + (350. + 254. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−313.+965.i)T+(−3.14e5−2.28e5i)T2 1 + (-313. + 965. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(898.−653.i)T+(1.52e5−4.68e5i)T2 1 + (898. - 653. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(51.6+37.5i)T+(1.76e5+5.43e5i)T2 1 + (51.6 + 37.5i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+286.T+7.04e5T2 1 + 286.T + 7.04e5T^{2}
97 1+(−1.40e3+1.02e3i)T+(2.82e5−8.68e5i)T2 1 + (-1.40e3 + 1.02e3i)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

−13.01193267990423189540588388921, −11.94002744289388030297442726763, −10.88062917687161907319962108392, −10.01766648027642211859740663603, −8.582208619162298612993883561935, −7.30194962789793316973895365166, −5.86205970168325635113806958967, −4.43446581997911994998657846088, −3.31186025463679516627737081860, −1.03061030836672758833808789726, 2.28477059614129082723958796945, 4.13245643412980690067736971254, 5.30105747326989967741202805519, 6.81478641552940006584448400217, 7.69918829350942413260746260853, 8.947388552277811839107734908864, 10.43798963940556020722775078793, 11.51956235615621018811642980621, 12.60888651477231969929215127720, 13.39683803828297146764994460147

Graph of the ZZ-function along the critical line