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Properties

Label 2-220-4.3-c2-0-39
Degree 22
Conductor 220220
Sign −0.00238−0.999i-0.00238 - 0.999i
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  + (−0.00238 − 1.99i)2-s − 5.13i·3-s + (−3.99 + 0.00955i)4-s − 2.23·5-s + (−10.2 + 0.0122i)6-s + 6.37i·7-s + (0.0286 + 7.99i)8-s − 17.3·9-s + (0.00533 + 4.47i)10-s − 3.31i·11-s + (0.0490 + 20.5i)12-s − 1.30·13-s + (12.7 − 0.0152i)14-s + 11.4i·15-s + (15.9 − 0.0764i)16-s − 24.9·17-s + ⋯
L(s)  = 1  + (−0.00119 − 0.999i)2-s − 1.71i·3-s + (−0.999 + 0.00238i)4-s − 0.447·5-s + (−1.71 + 0.00204i)6-s + 0.911i·7-s + (0.00358 + 0.999i)8-s − 1.92·9-s + (0.000533 + 0.447i)10-s − 0.301i·11-s + (0.00408 + 1.71i)12-s − 0.100·13-s + (0.911 − 0.00108i)14-s + 0.765i·15-s + (0.999 − 0.00477i)16-s − 1.46·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.00238−0.999i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00238 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.00238−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.00238−0.999i-0.00238 - 0.999i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(111,⋅)\chi_{220} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.00238−0.999i)(2,\ 220,\ (\ :1),\ -0.00238 - 0.999i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.359678+0.360538i0.359678 + 0.360538i
L(12)L(\frac12) ≈\approx 0.359678+0.360538i0.359678 + 0.360538i
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+(0.00238+1.99i)T 1 + (0.00238 + 1.99i)T
5 1+2.23T 1 + 2.23T
11 1+3.31iT 1 + 3.31iT
good3 1+5.13iT−9T2 1 + 5.13iT - 9T^{2}
7 1−6.37iT−49T2 1 - 6.37iT - 49T^{2}
13 1+1.30T+169T2 1 + 1.30T + 169T^{2}
17 1+24.9T+289T2 1 + 24.9T + 289T^{2}
19 1+29.5iT−361T2 1 + 29.5iT - 361T^{2}
23 1+4.05iT−529T2 1 + 4.05iT - 529T^{2}
29 1+6.14T+841T2 1 + 6.14T + 841T^{2}
31 1−16.6iT−961T2 1 - 16.6iT - 961T^{2}
37 1+11.3T+1.36e3T2 1 + 11.3T + 1.36e3T^{2}
41 1+46.1T+1.68e3T2 1 + 46.1T + 1.68e3T^{2}
43 1+51.0iT−1.84e3T2 1 + 51.0iT - 1.84e3T^{2}
47 1−74.9iT−2.20e3T2 1 - 74.9iT - 2.20e3T^{2}
53 1+52.6T+2.80e3T2 1 + 52.6T + 2.80e3T^{2}
59 1+11.6iT−3.48e3T2 1 + 11.6iT - 3.48e3T^{2}
61 1−62.5T+3.72e3T2 1 - 62.5T + 3.72e3T^{2}
67 1−49.7iT−4.48e3T2 1 - 49.7iT - 4.48e3T^{2}
71 1+132.iT−5.04e3T2 1 + 132. iT - 5.04e3T^{2}
73 1+113.T+5.32e3T2 1 + 113.T + 5.32e3T^{2}
79 1+144.iT−6.24e3T2 1 + 144. iT - 6.24e3T^{2}
83 1+67.6iT−6.88e3T2 1 + 67.6iT - 6.88e3T^{2}
89 1−155.T+7.92e3T2 1 - 155.T + 7.92e3T^{2}
97 1−163.T+9.40e3T2 1 - 163.T + 9.40e3T^{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

−11.61815384051976481170169391782, −10.83662427128112166005017928592, −9.007472313709219695901987548787, −8.581109872283463375857655090658, −7.34468208222787535723996621280, −6.23859186772818197063499473546, −4.86548648556205105436093575754, −2.94902107882452218126441279127, −1.95692677359588519435533378752, −0.27484814452698873027877878486, 3.70173936888732294773416110379, 4.29124969610334717819083466318, 5.30424949253546613414363734875, 6.66381905477249484998235442595, 7.917009508465619784516412199275, 8.860662321637006978316495303093, 9.873790091767117191972210169076, 10.46678357748583220625832211322, 11.58002733848919251756750290510, 13.04776186922666598459539412531

Graph of the ZZ-function along the critical line