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Properties

Label 2-220-5.3-c2-0-7
Degree 22
Conductor 220220
Sign −0.0983+0.995i-0.0983 + 0.995i
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  + (−1.52 − 1.52i)3-s + (4.49 − 2.19i)5-s + (0.0256 − 0.0256i)7-s − 4.34i·9-s − 3.31·11-s + (1.36 + 1.36i)13-s + (−10.1 − 3.49i)15-s + (4.75 − 4.75i)17-s − 24.5i·19-s − 0.0783·21-s + (−14.4 − 14.4i)23-s + (15.3 − 19.7i)25-s + (−20.3 + 20.3i)27-s − 51.3i·29-s − 8.20·31-s + ⋯
L(s)  = 1  + (−0.508 − 0.508i)3-s + (0.898 − 0.439i)5-s + (0.00367 − 0.00367i)7-s − 0.483i·9-s − 0.301·11-s + (0.105 + 0.105i)13-s + (−0.679 − 0.233i)15-s + (0.279 − 0.279i)17-s − 1.29i·19-s − 0.00373·21-s + (−0.627 − 0.627i)23-s + (0.613 − 0.789i)25-s + (−0.753 + 0.753i)27-s − 1.76i·29-s − 0.264·31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.0983+0.995i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.0983+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0983 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.0983+0.995i-0.0983 + 0.995i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(133,⋅)\chi_{220} (133, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.0983+0.995i)(2,\ 220,\ (\ :1),\ -0.0983 + 0.995i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.907864−1.00197i0.907864 - 1.00197i
L(12)L(\frac12) ≈\approx 0.907864−1.00197i0.907864 - 1.00197i
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+(−4.49+2.19i)T 1 + (-4.49 + 2.19i)T
11 1+3.31T 1 + 3.31T
good3 1+(1.52+1.52i)T+9iT2 1 + (1.52 + 1.52i)T + 9iT^{2}
7 1+(−0.0256+0.0256i)T−49iT2 1 + (-0.0256 + 0.0256i)T - 49iT^{2}
13 1+(−1.36−1.36i)T+169iT2 1 + (-1.36 - 1.36i)T + 169iT^{2}
17 1+(−4.75+4.75i)T−289iT2 1 + (-4.75 + 4.75i)T - 289iT^{2}
19 1+24.5iT−361T2 1 + 24.5iT - 361T^{2}
23 1+(14.4+14.4i)T+529iT2 1 + (14.4 + 14.4i)T + 529iT^{2}
29 1+51.3iT−841T2 1 + 51.3iT - 841T^{2}
31 1+8.20T+961T2 1 + 8.20T + 961T^{2}
37 1+(0.158−0.158i)T−1.36e3iT2 1 + (0.158 - 0.158i)T - 1.36e3iT^{2}
41 1−32.4T+1.68e3T2 1 - 32.4T + 1.68e3T^{2}
43 1+(−38.0−38.0i)T+1.84e3iT2 1 + (-38.0 - 38.0i)T + 1.84e3iT^{2}
47 1+(32.5−32.5i)T−2.20e3iT2 1 + (32.5 - 32.5i)T - 2.20e3iT^{2}
53 1+(−57.7−57.7i)T+2.80e3iT2 1 + (-57.7 - 57.7i)T + 2.80e3iT^{2}
59 1−2.32iT−3.48e3T2 1 - 2.32iT - 3.48e3T^{2}
61 1−34.8T+3.72e3T2 1 - 34.8T + 3.72e3T^{2}
67 1+(29.3−29.3i)T−4.48e3iT2 1 + (29.3 - 29.3i)T - 4.48e3iT^{2}
71 1−65.3T+5.04e3T2 1 - 65.3T + 5.04e3T^{2}
73 1+(29.3+29.3i)T+5.32e3iT2 1 + (29.3 + 29.3i)T + 5.32e3iT^{2}
79 1+50.5iT−6.24e3T2 1 + 50.5iT - 6.24e3T^{2}
83 1+(−72.6−72.6i)T+6.88e3iT2 1 + (-72.6 - 72.6i)T + 6.88e3iT^{2}
89 1+30.3iT−7.92e3T2 1 + 30.3iT - 7.92e3T^{2}
97 1+(34.2−34.2i)T−9.40e3iT2 1 + (34.2 - 34.2i)T - 9.40e3iT^{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

−11.91685270683465779001222471367, −10.93526355425301732399364033551, −9.748555472394354555200302024111, −9.013958846115846485426793405866, −7.68590943232058061648151946868, −6.45961205693873881823783514113, −5.75045294744345198633370488622, −4.45904295306416114250743502332, −2.50084754765512991761082270705, −0.812285033726378417554554389851, 1.94469228099144606545483090857, 3.63001622212411039576933039594, 5.23360077615689940995688962493, 5.85470060466772574468465048638, 7.18533298848092060924682240898, 8.399447383744524404558912467954, 9.737669495324787618614707479245, 10.37377392293097624841529610842, 11.08114696165531916234712100634, 12.28500413783974897185359889410

Graph of the ZZ-function along the critical line