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Properties

Label 2-220-11.7-c2-0-1
Degree 22
Conductor 220220
Sign −0.971−0.236i-0.971 - 0.236i
Analytic cond. 5.994565.99456
Root an. cond. 2.448382.44838
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 + 3.34i)3-s + (1.80 − 1.31i)5-s + (−6.54 + 2.12i)7-s + (−2.72 − 1.97i)9-s + (−1.85 + 10.8i)11-s + (−6.20 + 8.54i)13-s + (2.42 + 7.47i)15-s + (−14.5 − 19.9i)17-s + (−2.90 − 0.944i)19-s − 24.2i·21-s − 28.7·23-s + (1.54 − 4.75i)25-s + (−16.0 + 11.6i)27-s + (−14.4 + 4.70i)29-s + (22.8 + 16.6i)31-s + ⋯
L(s)  = 1  + (−0.362 + 1.11i)3-s + (0.361 − 0.262i)5-s + (−0.935 + 0.303i)7-s + (−0.302 − 0.219i)9-s + (−0.168 + 0.985i)11-s + (−0.477 + 0.657i)13-s + (0.161 + 0.498i)15-s + (−0.853 − 1.17i)17-s + (−0.153 − 0.0497i)19-s − 1.15i·21-s − 1.25·23-s + (0.0618 − 0.190i)25-s + (−0.593 + 0.431i)27-s + (−0.499 + 0.162i)29-s + (0.738 + 0.536i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.971−0.236i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.971−0.236i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.971−0.236i-0.971 - 0.236i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(161,⋅)\chi_{220} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.971−0.236i)(2,\ 220,\ (\ :1),\ -0.971 - 0.236i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0899299+0.748364i0.0899299 + 0.748364i
L(12)L(\frac12) ≈\approx 0.0899299+0.748364i0.0899299 + 0.748364i
L(2)L(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
5 1+(−1.80+1.31i)T 1 + (-1.80 + 1.31i)T
11 1+(1.85−10.8i)T 1 + (1.85 - 10.8i)T
good3 1+(1.08−3.34i)T+(−7.28−5.29i)T2 1 + (1.08 - 3.34i)T + (-7.28 - 5.29i)T^{2}
7 1+(6.54−2.12i)T+(39.6−28.8i)T2 1 + (6.54 - 2.12i)T + (39.6 - 28.8i)T^{2}
13 1+(6.20−8.54i)T+(−52.2−160.i)T2 1 + (6.20 - 8.54i)T + (-52.2 - 160. i)T^{2}
17 1+(14.5+19.9i)T+(−89.3+274.i)T2 1 + (14.5 + 19.9i)T + (-89.3 + 274. i)T^{2}
19 1+(2.90+0.944i)T+(292.+212.i)T2 1 + (2.90 + 0.944i)T + (292. + 212. i)T^{2}
23 1+28.7T+529T2 1 + 28.7T + 529T^{2}
29 1+(14.4−4.70i)T+(680.−494.i)T2 1 + (14.4 - 4.70i)T + (680. - 494. i)T^{2}
31 1+(−22.8−16.6i)T+(296.+913.i)T2 1 + (-22.8 - 16.6i)T + (296. + 913. i)T^{2}
37 1+(0.847+2.60i)T+(−1.10e3+804.i)T2 1 + (0.847 + 2.60i)T + (-1.10e3 + 804. i)T^{2}
41 1+(−58.6−19.0i)T+(1.35e3+988.i)T2 1 + (-58.6 - 19.0i)T + (1.35e3 + 988. i)T^{2}
43 1−40.5iT−1.84e3T2 1 - 40.5iT - 1.84e3T^{2}
47 1+(−7.00+21.5i)T+(−1.78e3−1.29e3i)T2 1 + (-7.00 + 21.5i)T + (-1.78e3 - 1.29e3i)T^{2}
53 1+(−69.9−50.8i)T+(868.+2.67e3i)T2 1 + (-69.9 - 50.8i)T + (868. + 2.67e3i)T^{2}
59 1+(−22.7−70.1i)T+(−2.81e3+2.04e3i)T2 1 + (-22.7 - 70.1i)T + (-2.81e3 + 2.04e3i)T^{2}
61 1+(34.2+47.1i)T+(−1.14e3+3.53e3i)T2 1 + (34.2 + 47.1i)T + (-1.14e3 + 3.53e3i)T^{2}
67 1−64.6T+4.48e3T2 1 - 64.6T + 4.48e3T^{2}
71 1+(−49.4+35.9i)T+(1.55e3−4.79e3i)T2 1 + (-49.4 + 35.9i)T + (1.55e3 - 4.79e3i)T^{2}
73 1+(80.2−26.0i)T+(4.31e3−3.13e3i)T2 1 + (80.2 - 26.0i)T + (4.31e3 - 3.13e3i)T^{2}
79 1+(44.8−61.7i)T+(−1.92e3−5.93e3i)T2 1 + (44.8 - 61.7i)T + (-1.92e3 - 5.93e3i)T^{2}
83 1+(11.1+15.2i)T+(−2.12e3+6.55e3i)T2 1 + (11.1 + 15.2i)T + (-2.12e3 + 6.55e3i)T^{2}
89 1−127.T+7.92e3T2 1 - 127.T + 7.92e3T^{2}
97 1+(−71.6−52.0i)T+(2.90e3+8.94e3i)T2 1 + (-71.6 - 52.0i)T + (2.90e3 + 8.94e3i)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

−12.42379773502182527248776974467, −11.50039546796559675203740811348, −10.30036224972609564783318741150, −9.639728843558179518731744330736, −9.077763667478264109541978473134, −7.36560450364445617277856966645, −6.22288993831659602584989354865, −4.97930615329424284221507119874, −4.17451400514341963007251869338, −2.44038482446670441936589880759, 0.40749603546298301948403356511, 2.24832094433324393884940285935, 3.80656936747260255670372073061, 5.83968630792547546966490594910, 6.35097026063500771315932683938, 7.41741293294075849502207595229, 8.445462617608429071025704391246, 9.819338775015604086925921171295, 10.65488336558283956703400403032, 11.75972917665926412346251118152

Graph of the ZZ-function along the critical line