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Properties

Label 2-110-11.5-c3-0-11
Degree 22
Conductor 110110
Sign −0.531+0.846i-0.531 + 0.846i
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 + (−2.02 − 1.46i)3-s + (−3.23 + 2.35i)4-s + (1.54 − 4.75i)5-s + (1.54 − 4.75i)6-s + (−14.4 + 10.4i)7-s + (−6.47 − 4.70i)8-s + (−6.41 − 19.7i)9-s + 10.0·10-s + (−36.2 + 4.12i)11-s + 9.99·12-s + (−12.0 − 37.1i)13-s + (−28.8 − 20.9i)14-s + (−10.1 + 7.34i)15-s + (4.94 − 15.2i)16-s + (3.94 − 12.1i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.389 − 0.282i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (0.105 − 0.323i)6-s + (−0.777 + 0.565i)7-s + (−0.286 − 0.207i)8-s + (−0.237 − 0.731i)9-s + 0.316·10-s + (−0.993 + 0.113i)11-s + 0.240·12-s + (−0.257 − 0.792i)13-s + (−0.550 − 0.399i)14-s + (−0.174 + 0.126i)15-s + (0.0772 − 0.237i)16-s + (0.0562 − 0.173i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.531+0.846i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+3/2) L(s)=((−0.531+0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.531+0.846i-0.531 + 0.846i
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), −0.531+0.846i)(2,\ 110,\ (\ :3/2),\ -0.531 + 0.846i)

Particular Values

L(2)L(2) ≈\approx 0.174879−0.316321i0.174879 - 0.316321i
L(12)L(\frac12) ≈\approx 0.174879−0.316321i0.174879 - 0.316321i
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+(36.2−4.12i)T 1 + (36.2 - 4.12i)T
good3 1+(2.02+1.46i)T+(8.34+25.6i)T2 1 + (2.02 + 1.46i)T + (8.34 + 25.6i)T^{2}
7 1+(14.4−10.4i)T+(105.−326.i)T2 1 + (14.4 - 10.4i)T + (105. - 326. i)T^{2}
13 1+(12.0+37.1i)T+(−1.77e3+1.29e3i)T2 1 + (12.0 + 37.1i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(−3.94+12.1i)T+(−3.97e3−2.88e3i)T2 1 + (-3.94 + 12.1i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(95.3+69.2i)T+(2.11e3+6.52e3i)T2 1 + (95.3 + 69.2i)T + (2.11e3 + 6.52e3i)T^{2}
23 1−62.4T+1.21e4T2 1 - 62.4T + 1.21e4T^{2}
29 1+(−11.0+8.03i)T+(7.53e3−2.31e4i)T2 1 + (-11.0 + 8.03i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(−49.7−153.i)T+(−2.41e4+1.75e4i)T2 1 + (-49.7 - 153. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(182.−132.i)T+(1.56e4−4.81e4i)T2 1 + (182. - 132. i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(213.+155.i)T+(2.12e4+6.55e4i)T2 1 + (213. + 155. i)T + (2.12e4 + 6.55e4i)T^{2}
43 1−154.T+7.95e4T2 1 - 154.T + 7.95e4T^{2}
47 1+(−126.−92.0i)T+(3.20e4+9.87e4i)T2 1 + (-126. - 92.0i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(−54.3−167.i)T+(−1.20e5+8.75e4i)T2 1 + (-54.3 - 167. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(701.−509.i)T+(6.34e4−1.95e5i)T2 1 + (701. - 509. i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(−95.0+292.i)T+(−1.83e5−1.33e5i)T2 1 + (-95.0 + 292. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1−732.T+3.00e5T2 1 - 732.T + 3.00e5T^{2}
71 1+(−203.+625.i)T+(−2.89e5−2.10e5i)T2 1 + (-203. + 625. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−976.+709.i)T+(1.20e5−3.69e5i)T2 1 + (-976. + 709. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(252.+778.i)T+(−3.98e5+2.89e5i)T2 1 + (252. + 778. i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(53.8−165.i)T+(−4.62e5−3.36e5i)T2 1 + (53.8 - 165. i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1+892.T+7.04e5T2 1 + 892.T + 7.04e5T^{2}
97 1+(445.+1.37e3i)T+(−7.38e5+5.36e5i)T2 1 + (445. + 1.37e3i)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

−12.69388453416269476373842427389, −12.32507239515825516412340926990, −10.69612926696695795792968019832, −9.376530640224620879369564155234, −8.428632926489387905859255218695, −7.00349815906581743971096265381, −5.97746766114428535195758478540, −4.96141183712938516336151297434, −3.00006574062740258294889728806, −0.18082874822178170785966694957, 2.37179088090313240999662750624, 3.94836016755950988878156257895, 5.33434993068681128842672018524, 6.65677820154571717266549949344, 8.159959547993356244860913398920, 9.736007700204023344231985294701, 10.50780834918155704274171032604, 11.20632009110183607615641926031, 12.54564138473846904785891021287, 13.41949613368273002538774977797

Graph of the ZZ-function along the critical line