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Properties

Label 2-110-11.8-c4-0-10
Degree 22
Conductor 110110
Sign −0.516+0.855i-0.516 + 0.855i
Analytic cond. 11.370611.3706
Root an. cond. 3.372043.37204
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.66 − 2.28i)2-s + (−2.06 − 6.36i)3-s + (−2.47 + 7.60i)4-s + (−9.04 − 6.57i)5-s + (−11.1 + 15.3i)6-s + (87.9 + 28.5i)7-s + (21.5 − 6.99i)8-s + (29.3 − 21.2i)9-s + 31.6i·10-s + (114. − 39.1i)11-s + 53.5·12-s + (−177. − 244. i)13-s + (−80.8 − 248. i)14-s + (−23.1 + 71.1i)15-s + (−51.7 − 37.6i)16-s + (−235. + 324. i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.229 − 0.707i)3-s + (−0.154 + 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.308 + 0.425i)6-s + (1.79 + 0.583i)7-s + (0.336 − 0.109i)8-s + (0.361 − 0.262i)9-s + 0.316i·10-s + (0.946 − 0.323i)11-s + 0.371·12-s + (−1.05 − 1.44i)13-s + (−0.412 − 1.26i)14-s + (−0.102 + 0.316i)15-s + (−0.202 − 0.146i)16-s + (−0.815 + 1.12i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.516+0.855i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+2) L(s)=((−0.516+0.855i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.516+0.855i-0.516 + 0.855i
Analytic conductor: 11.370611.3706
Root analytic conductor: 3.372043.37204
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ110(41,⋅)\chi_{110} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :2), −0.516+0.855i)(2,\ 110,\ (\ :2),\ -0.516 + 0.855i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.688434−1.22002i0.688434 - 1.22002i
L(12)L(\frac12) ≈\approx 0.688434−1.22002i0.688434 - 1.22002i
L(3)L(3) 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+(1.66+2.28i)T 1 + (1.66 + 2.28i)T
5 1+(9.04+6.57i)T 1 + (9.04 + 6.57i)T
11 1+(−114.+39.1i)T 1 + (-114. + 39.1i)T
good3 1+(2.06+6.36i)T+(−65.5+47.6i)T2 1 + (2.06 + 6.36i)T + (-65.5 + 47.6i)T^{2}
7 1+(−87.9−28.5i)T+(1.94e3+1.41e3i)T2 1 + (-87.9 - 28.5i)T + (1.94e3 + 1.41e3i)T^{2}
13 1+(177.+244.i)T+(−8.82e3+2.71e4i)T2 1 + (177. + 244. i)T + (-8.82e3 + 2.71e4i)T^{2}
17 1+(235.−324.i)T+(−2.58e4−7.94e4i)T2 1 + (235. - 324. i)T + (-2.58e4 - 7.94e4i)T^{2}
19 1+(−272.+88.4i)T+(1.05e5−7.66e4i)T2 1 + (-272. + 88.4i)T + (1.05e5 - 7.66e4i)T^{2}
23 1−73.4T+2.79e5T2 1 - 73.4T + 2.79e5T^{2}
29 1+(1.04e3+338.i)T+(5.72e5+4.15e5i)T2 1 + (1.04e3 + 338. i)T + (5.72e5 + 4.15e5i)T^{2}
31 1+(−878.+637.i)T+(2.85e5−8.78e5i)T2 1 + (-878. + 637. i)T + (2.85e5 - 8.78e5i)T^{2}
37 1+(−677.+2.08e3i)T+(−1.51e6−1.10e6i)T2 1 + (-677. + 2.08e3i)T + (-1.51e6 - 1.10e6i)T^{2}
41 1+(−216.+70.4i)T+(2.28e6−1.66e6i)T2 1 + (-216. + 70.4i)T + (2.28e6 - 1.66e6i)T^{2}
43 1+1.76e3iT−3.41e6T2 1 + 1.76e3iT - 3.41e6T^{2}
47 1+(−19.7−60.7i)T+(−3.94e6+2.86e6i)T2 1 + (-19.7 - 60.7i)T + (-3.94e6 + 2.86e6i)T^{2}
53 1+(−1.40e3+1.02e3i)T+(2.43e6−7.50e6i)T2 1 + (-1.40e3 + 1.02e3i)T + (2.43e6 - 7.50e6i)T^{2}
59 1+(1.10e3−3.40e3i)T+(−9.80e6−7.12e6i)T2 1 + (1.10e3 - 3.40e3i)T + (-9.80e6 - 7.12e6i)T^{2}
61 1+(−278.+383.i)T+(−4.27e6−1.31e7i)T2 1 + (-278. + 383. i)T + (-4.27e6 - 1.31e7i)T^{2}
67 1+3.53e3T+2.01e7T2 1 + 3.53e3T + 2.01e7T^{2}
71 1+(2.19e3+1.59e3i)T+(7.85e6+2.41e7i)T2 1 + (2.19e3 + 1.59e3i)T + (7.85e6 + 2.41e7i)T^{2}
73 1+(−1.25e3−407.i)T+(2.29e7+1.66e7i)T2 1 + (-1.25e3 - 407. i)T + (2.29e7 + 1.66e7i)T^{2}
79 1+(−1.91e3−2.64e3i)T+(−1.20e7+3.70e7i)T2 1 + (-1.91e3 - 2.64e3i)T + (-1.20e7 + 3.70e7i)T^{2}
83 1+(4.25e3−5.85e3i)T+(−1.46e7−4.51e7i)T2 1 + (4.25e3 - 5.85e3i)T + (-1.46e7 - 4.51e7i)T^{2}
89 1−8.32e3T+6.27e7T2 1 - 8.32e3T + 6.27e7T^{2}
97 1+(9.42e3−6.84e3i)T+(2.73e7−8.41e7i)T2 1 + (9.42e3 - 6.84e3i)T + (2.73e7 - 8.41e7i)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

−12.28236934611918057823736418614, −11.69279698253942834218241108917, −10.76235963583766164588765505114, −9.254117671373915087266064764037, −8.150184333979090330103245336502, −7.41653534293237236273069740729, −5.61796500738436354600000654344, −4.18068510439725967314007179480, −2.08371237223992447318529394573, −0.822057650508988985183642841846, 1.59532196937924567677000766009, 4.43581411322666052005925493145, 4.83548296885355214511321966782, 6.95026413944093817794115781636, 7.62689504431634659260887257455, 9.057217776223549987851749745155, 10.00062509681836220255419941825, 11.28597306744895776123871996150, 11.68895636385256193277170580730, 13.81313549413059872937068157471

Graph of the ZZ-function along the critical line