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Properties

Label 2-110-11.3-c3-0-2
Degree 22
Conductor 110110
Sign 0.967−0.254i0.967 - 0.254i
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 + (−0.678 − 2.08i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (3.55 + 2.57i)6-s + (−10.1 + 31.2i)7-s + (2.47 + 7.60i)8-s + (17.9 − 13.0i)9-s + 10·10-s + (22.4 − 28.7i)11-s − 8.77·12-s + (52.1 − 37.9i)13-s + (−20.2 − 62.4i)14-s + (−3.39 + 10.4i)15-s + (−12.9 − 9.40i)16-s + (77.1 + 56.0i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.130 − 0.401i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.241 + 0.175i)6-s + (−0.548 + 1.68i)7-s + (0.109 + 0.336i)8-s + (0.664 − 0.482i)9-s + 0.316·10-s + (0.614 − 0.789i)11-s − 0.211·12-s + (1.11 − 0.808i)13-s + (−0.387 − 1.19i)14-s + (−0.0583 + 0.179i)15-s + (−0.202 − 0.146i)16-s + (1.10 + 0.799i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((0.967−0.254i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((0.967−0.254i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: 0.967−0.254i0.967 - 0.254i
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), 0.967−0.254i)(2,\ 110,\ (\ :3/2),\ 0.967 - 0.254i)

Particular Values

L(2)L(2) ≈\approx 1.14394+0.147990i1.14394 + 0.147990i
L(12)L(\frac12) ≈\approx 1.14394+0.147990i1.14394 + 0.147990i
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+(−22.4+28.7i)T 1 + (-22.4 + 28.7i)T
good3 1+(0.678+2.08i)T+(−21.8+15.8i)T2 1 + (0.678 + 2.08i)T + (-21.8 + 15.8i)T^{2}
7 1+(10.1−31.2i)T+(−277.−201.i)T2 1 + (10.1 - 31.2i)T + (-277. - 201. i)T^{2}
13 1+(−52.1+37.9i)T+(678.−2.08e3i)T2 1 + (-52.1 + 37.9i)T + (678. - 2.08e3i)T^{2}
17 1+(−77.1−56.0i)T+(1.51e3+4.67e3i)T2 1 + (-77.1 - 56.0i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(−21.2−65.5i)T+(−5.54e3+4.03e3i)T2 1 + (-21.2 - 65.5i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−112.T+1.21e4T2 1 - 112.T + 1.21e4T^{2}
29 1+(−18.1+55.9i)T+(−1.97e4−1.43e4i)T2 1 + (-18.1 + 55.9i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(6.31−4.58i)T+(9.20e3−2.83e4i)T2 1 + (6.31 - 4.58i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−34.2+105.i)T+(−4.09e4−2.97e4i)T2 1 + (-34.2 + 105. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−42.2−130.i)T+(−5.57e4+4.05e4i)T2 1 + (-42.2 - 130. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1+23.7T+7.95e4T2 1 + 23.7T + 7.95e4T^{2}
47 1+(−167.−516.i)T+(−8.39e4+6.10e4i)T2 1 + (-167. - 516. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(38.8−28.2i)T+(4.60e4−1.41e5i)T2 1 + (38.8 - 28.2i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−272.+838.i)T+(−1.66e5−1.20e5i)T2 1 + (-272. + 838. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(339.+246.i)T+(7.01e4+2.15e5i)T2 1 + (339. + 246. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1+726.T+3.00e5T2 1 + 726.T + 3.00e5T^{2}
71 1+(−921.−669.i)T+(1.10e5+3.40e5i)T2 1 + (-921. - 669. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(41.9−129.i)T+(−3.14e5−2.28e5i)T2 1 + (41.9 - 129. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(333.−242.i)T+(1.52e5−4.68e5i)T2 1 + (333. - 242. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(110.+80.0i)T+(1.76e5+5.43e5i)T2 1 + (110. + 80.0i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+492.T+7.04e5T2 1 + 492.T + 7.04e5T^{2}
97 1+(−997.+724.i)T+(2.82e5−8.68e5i)T2 1 + (-997. + 724. 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

−12.91893991173974483541195756400, −12.29753368525829730120115584158, −11.19668386422021428371443663918, −9.735741645913600794333095673565, −8.782516299042087027574010116644, −7.930249942623643887564334350281, −6.30124994075559795456040057725, −5.70283208974898972650823755862, −3.39828311370164772762037430070, −1.15582733606615493691160718329, 1.10419481947305048895514805951, 3.52294502204010396492092696932, 4.50614143513939556355980674303, 6.92303447521141865557182081931, 7.39261507406241713899196303270, 9.133533383915780969319828956146, 10.10509277792402678731081168656, 10.80608739340610524252432969907, 11.80993739387766166673700938474, 13.18747695674494460112669370024

Graph of the ZZ-function along the critical line