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Properties

Label 2-220-20.3-c1-0-5
Degree 22
Conductor 220220
Sign 0.976−0.213i0.976 - 0.213i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 − 0.857i)2-s + (0.635 − 0.635i)3-s + (0.530 + 1.92i)4-s + (−1.96 + 1.07i)5-s + (−1.25 + 0.170i)6-s + (2.25 + 2.25i)7-s + (1.05 − 2.62i)8-s + 2.19i·9-s + (3.12 + 0.475i)10-s − i·11-s + (1.56 + 0.887i)12-s + (3.81 + 3.81i)13-s + (−0.603 − 4.46i)14-s + (−0.565 + 1.92i)15-s + (−3.43 + 2.04i)16-s + (2.04 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.795 − 0.606i)2-s + (0.366 − 0.366i)3-s + (0.265 + 0.964i)4-s + (−0.877 + 0.479i)5-s + (−0.513 + 0.0694i)6-s + (0.851 + 0.851i)7-s + (0.373 − 0.927i)8-s + 0.731i·9-s + (0.988 + 0.150i)10-s − 0.301i·11-s + (0.450 + 0.256i)12-s + (1.05 + 1.05i)13-s + (−0.161 − 1.19i)14-s + (−0.145 + 0.497i)15-s + (−0.859 + 0.511i)16-s + (0.495 − 0.495i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.976−0.213i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.976−0.213i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.976−0.213i0.976 - 0.213i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(23,⋅)\chi_{220} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.976−0.213i)(2,\ 220,\ (\ :1/2),\ 0.976 - 0.213i)

Particular Values

L(1)L(1) ≈\approx 0.886623+0.0957328i0.886623 + 0.0957328i
L(12)L(\frac12) ≈\approx 0.886623+0.0957328i0.886623 + 0.0957328i
L(32)L(\frac{3}{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+(1.12+0.857i)T 1 + (1.12 + 0.857i)T
5 1+(1.96−1.07i)T 1 + (1.96 - 1.07i)T
11 1+iT 1 + iT
good3 1+(−0.635+0.635i)T−3iT2 1 + (-0.635 + 0.635i)T - 3iT^{2}
7 1+(−2.25−2.25i)T+7iT2 1 + (-2.25 - 2.25i)T + 7iT^{2}
13 1+(−3.81−3.81i)T+13iT2 1 + (-3.81 - 3.81i)T + 13iT^{2}
17 1+(−2.04+2.04i)T−17iT2 1 + (-2.04 + 2.04i)T - 17iT^{2}
19 1+3.69T+19T2 1 + 3.69T + 19T^{2}
23 1+(−2.52+2.52i)T−23iT2 1 + (-2.52 + 2.52i)T - 23iT^{2}
29 1−8.82iT−29T2 1 - 8.82iT - 29T^{2}
31 1+6.21iT−31T2 1 + 6.21iT - 31T^{2}
37 1+(−1.82+1.82i)T−37iT2 1 + (-1.82 + 1.82i)T - 37iT^{2}
41 1+0.00765T+41T2 1 + 0.00765T + 41T^{2}
43 1+(5.73−5.73i)T−43iT2 1 + (5.73 - 5.73i)T - 43iT^{2}
47 1+(−1.19−1.19i)T+47iT2 1 + (-1.19 - 1.19i)T + 47iT^{2}
53 1+(5.59+5.59i)T+53iT2 1 + (5.59 + 5.59i)T + 53iT^{2}
59 1+14.2T+59T2 1 + 14.2T + 59T^{2}
61 1−2.14T+61T2 1 - 2.14T + 61T^{2}
67 1+(3.37+3.37i)T+67iT2 1 + (3.37 + 3.37i)T + 67iT^{2}
71 1+0.207iT−71T2 1 + 0.207iT - 71T^{2}
73 1+(0.0410+0.0410i)T+73iT2 1 + (0.0410 + 0.0410i)T + 73iT^{2}
79 1−11.2T+79T2 1 - 11.2T + 79T^{2}
83 1+(−11.8+11.8i)T−83iT2 1 + (-11.8 + 11.8i)T - 83iT^{2}
89 1−5.49iT−89T2 1 - 5.49iT - 89T^{2}
97 1+(−8.27+8.27i)T−97iT2 1 + (-8.27 + 8.27i)T - 97iT^{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

−12.02248588291314398797118836801, −11.19880688046358960369235286897, −10.76188391536327613277297343353, −9.098041084073446700339313952413, −8.398126205458740455663616431974, −7.70795912472549661335245034077, −6.56408103595674591112343276201, −4.61341756862519114578008087879, −3.14361094563429899058703344062, −1.83474845829345656917256561361, 1.06490985676965274702118492390, 3.67716448612102021369433152199, 4.83341439626734801106502094567, 6.27953215998650740132984360598, 7.62879238016649699543432616368, 8.215303808750242256033883211758, 9.049915044097271451614702847427, 10.29511173174802468267905709241, 10.97157635053633939103320843745, 12.06024905126489751375539840843

Graph of the ZZ-function along the critical line