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Properties

Label 2-110-11.5-c3-0-10
Degree 22
Conductor 110110
Sign −0.746+0.665i-0.746 + 0.665i
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  + (−0.618 − 1.90i)2-s + (−0.396 − 0.287i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (−0.302 + 0.931i)6-s + (19.1 − 13.9i)7-s + (6.47 + 4.70i)8-s + (−8.26 − 25.4i)9-s + 10.0·10-s + (−35.9 − 6.11i)11-s + 1.95·12-s + (−22.0 − 67.9i)13-s + (−38.3 − 27.8i)14-s + (1.98 − 1.43i)15-s + (4.94 − 15.2i)16-s + (−8.55 + 26.3i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.0762 − 0.0554i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.0206 + 0.0634i)6-s + (1.03 − 0.751i)7-s + (0.286 + 0.207i)8-s + (−0.306 − 0.942i)9-s + 0.316·10-s + (−0.985 − 0.167i)11-s + 0.0471·12-s + (−0.470 − 1.44i)13-s + (−0.731 − 0.531i)14-s + (0.0341 − 0.0247i)15-s + (0.0772 − 0.237i)16-s + (−0.122 + 0.375i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.746+0.665i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.746+0.665i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.746+0.665i-0.746 + 0.665i
Analytic conductor: 6.490216.49021
Root analytic conductor: 2.547582.54758
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ110(71,⋅)\chi_{110} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :3/2), −0.746+0.665i)(2,\ 110,\ (\ :3/2),\ -0.746 + 0.665i)

Particular Values

L(2)L(2) ≈\approx 0.362357−0.950530i0.362357 - 0.950530i
L(12)L(\frac12) ≈\approx 0.362357−0.950530i0.362357 - 0.950530i
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+(0.618+1.90i)T 1 + (0.618 + 1.90i)T
5 1+(1.54−4.75i)T 1 + (1.54 - 4.75i)T
11 1+(35.9+6.11i)T 1 + (35.9 + 6.11i)T
good3 1+(0.396+0.287i)T+(8.34+25.6i)T2 1 + (0.396 + 0.287i)T + (8.34 + 25.6i)T^{2}
7 1+(−19.1+13.9i)T+(105.−326.i)T2 1 + (-19.1 + 13.9i)T + (105. - 326. i)T^{2}
13 1+(22.0+67.9i)T+(−1.77e3+1.29e3i)T2 1 + (22.0 + 67.9i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(8.55−26.3i)T+(−3.97e3−2.88e3i)T2 1 + (8.55 - 26.3i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(−3.30−2.39i)T+(2.11e3+6.52e3i)T2 1 + (-3.30 - 2.39i)T + (2.11e3 + 6.52e3i)T^{2}
23 1+74.7T+1.21e4T2 1 + 74.7T + 1.21e4T^{2}
29 1+(−229.+166.i)T+(7.53e3−2.31e4i)T2 1 + (-229. + 166. i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(92.0+283.i)T+(−2.41e4+1.75e4i)T2 1 + (92.0 + 283. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(89.3−64.9i)T+(1.56e4−4.81e4i)T2 1 + (89.3 - 64.9i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(−151.−110.i)T+(2.12e4+6.55e4i)T2 1 + (-151. - 110. i)T + (2.12e4 + 6.55e4i)T^{2}
43 1−131.T+7.95e4T2 1 - 131.T + 7.95e4T^{2}
47 1+(−97.8−71.0i)T+(3.20e4+9.87e4i)T2 1 + (-97.8 - 71.0i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(−165.−509.i)T+(−1.20e5+8.75e4i)T2 1 + (-165. - 509. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(−421.+305.i)T+(6.34e4−1.95e5i)T2 1 + (-421. + 305. i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(216.−665.i)T+(−1.83e5−1.33e5i)T2 1 + (216. - 665. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1−710.T+3.00e5T2 1 - 710.T + 3.00e5T^{2}
71 1+(−62.2+191.i)T+(−2.89e5−2.10e5i)T2 1 + (-62.2 + 191. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(458.−332.i)T+(1.20e5−3.69e5i)T2 1 + (458. - 332. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(349.+1.07e3i)T+(−3.98e5+2.89e5i)T2 1 + (349. + 1.07e3i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(444.−1.36e3i)T+(−4.62e5−3.36e5i)T2 1 + (444. - 1.36e3i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1−575.T+7.04e5T2 1 - 575.T + 7.04e5T^{2}
97 1+(255.+785.i)T+(−7.38e5+5.36e5i)T2 1 + (255. + 785. i)T + (-7.38e5 + 5.36e5i)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

−12.64165922808025300088317993018, −11.59387261217012300432387729109, −10.67755385066721554271095149769, −9.929608732158537628102839128171, −8.272864725868494467391013401880, −7.57849576713423574970466174922, −5.77313484074140804684990930249, −4.21624195450614836228712778887, −2.71175328366351753105260510423, −0.60299120426095129596649922322, 2.07568525085981082178266969251, 4.74203248781539095572739966582, 5.33351224649979611190821190633, 7.07685724392759830260844392764, 8.214515329238273234440850507935, 8.924264269346698547870547359014, 10.37017991438765200337213060560, 11.49937097666952748167497350257, 12.52053127123546729443856330192, 13.95059152564324384444796271436

Graph of the ZZ-function along the critical line