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Properties

Label 2-220-4.3-c2-0-5
Degree 22
Conductor 220220
Sign −0.638+0.769i-0.638 + 0.769i
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  + (−0.679 + 1.88i)2-s + 5.18i·3-s + (−3.07 − 2.55i)4-s + 2.23·5-s + (−9.75 − 3.52i)6-s + 11.9i·7-s + (6.89 − 4.05i)8-s − 17.8·9-s + (−1.51 + 4.20i)10-s − 3.31i·11-s + (13.2 − 15.9i)12-s − 14.0·13-s + (−22.5 − 8.13i)14-s + 11.5i·15-s + (2.94 + 15.7i)16-s + 20.7·17-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)2-s + 1.72i·3-s + (−0.769 − 0.638i)4-s + 0.447·5-s + (−1.62 − 0.587i)6-s + 1.71i·7-s + (0.862 − 0.506i)8-s − 1.98·9-s + (−0.151 + 0.420i)10-s − 0.301i·11-s + (1.10 − 1.32i)12-s − 1.08·13-s + (−1.60 − 0.581i)14-s + 0.773i·15-s + (0.183 + 0.982i)16-s + 1.22·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.638+0.769i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.638+0.769i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.638+0.769i-0.638 + 0.769i
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), −0.638+0.769i)(2,\ 220,\ (\ :1),\ -0.638 + 0.769i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.442972−0.943621i0.442972 - 0.943621i
L(12)L(\frac12) ≈\approx 0.442972−0.943621i0.442972 - 0.943621i
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+(0.679−1.88i)T 1 + (0.679 - 1.88i)T
5 1−2.23T 1 - 2.23T
11 1+3.31iT 1 + 3.31iT
good3 1−5.18iT−9T2 1 - 5.18iT - 9T^{2}
7 1−11.9iT−49T2 1 - 11.9iT - 49T^{2}
13 1+14.0T+169T2 1 + 14.0T + 169T^{2}
17 1−20.7T+289T2 1 - 20.7T + 289T^{2}
19 1−0.389iT−361T2 1 - 0.389iT - 361T^{2}
23 1+7.30iT−529T2 1 + 7.30iT - 529T^{2}
29 1−39.6T+841T2 1 - 39.6T + 841T^{2}
31 1+24.7iT−961T2 1 + 24.7iT - 961T^{2}
37 1−28.5T+1.36e3T2 1 - 28.5T + 1.36e3T^{2}
41 1+41.8T+1.68e3T2 1 + 41.8T + 1.68e3T^{2}
43 1+2.07iT−1.84e3T2 1 + 2.07iT - 1.84e3T^{2}
47 1−87.2iT−2.20e3T2 1 - 87.2iT - 2.20e3T^{2}
53 1+103.T+2.80e3T2 1 + 103.T + 2.80e3T^{2}
59 1−44.6iT−3.48e3T2 1 - 44.6iT - 3.48e3T^{2}
61 1−61.7T+3.72e3T2 1 - 61.7T + 3.72e3T^{2}
67 1−36.1iT−4.48e3T2 1 - 36.1iT - 4.48e3T^{2}
71 1−98.8iT−5.04e3T2 1 - 98.8iT - 5.04e3T^{2}
73 1+10.3T+5.32e3T2 1 + 10.3T + 5.32e3T^{2}
79 1+59.9iT−6.24e3T2 1 + 59.9iT - 6.24e3T^{2}
83 1−23.4iT−6.88e3T2 1 - 23.4iT - 6.88e3T^{2}
89 1+55.8T+7.92e3T2 1 + 55.8T + 7.92e3T^{2}
97 1−86.7T+9.40e3T2 1 - 86.7T + 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

−12.64093458875625216282069529380, −11.51836812371565218990386272068, −10.18907421660070387117111027680, −9.662306464215124713559059575501, −8.937104592009706986209676693936, −8.044659853525609717542017005903, −6.15173150188596364990073198861, −5.40984931305619557418815040539, −4.62967854185190857137880263100, −2.87078879205141979280058548253, 0.66518569454909269345638266438, 1.75134699295395282826293209549, 3.21219150045635339160283738689, 4.94257319933837260730487973717, 6.72302998230954886110955283338, 7.48619615056692915309294567427, 8.202467614637206747693751705057, 9.756149210580271326778245454639, 10.45497571429484728704599616090, 11.67128569091152390441299044241

Graph of the ZZ-function along the critical line