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Properties

Label 2-220-44.43-c1-0-10
Degree 22
Conductor 220220
Sign 0.343+0.939i0.343 + 0.939i
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  + (−0.960 − 1.03i)2-s − 0.450i·3-s + (−0.154 + 1.99i)4-s + 5-s + (−0.467 + 0.432i)6-s − 1.25·7-s + (2.21 − 1.75i)8-s + 2.79·9-s + (−0.960 − 1.03i)10-s + (3.01 − 1.37i)11-s + (0.898 + 0.0698i)12-s − 3.65i·13-s + (1.20 + 1.30i)14-s − 0.450i·15-s + (−3.95 − 0.618i)16-s + 1.35i·17-s + ⋯
L(s)  = 1  + (−0.679 − 0.733i)2-s − 0.260i·3-s + (−0.0774 + 0.996i)4-s + 0.447·5-s + (−0.190 + 0.176i)6-s − 0.475·7-s + (0.784 − 0.620i)8-s + 0.932·9-s + (−0.303 − 0.328i)10-s + (0.909 − 0.414i)11-s + (0.259 + 0.0201i)12-s − 1.01i·13-s + (0.322 + 0.349i)14-s − 0.116i·15-s + (−0.987 − 0.154i)16-s + 0.329i·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.343+0.939i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.343+0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.343+0.939i0.343 + 0.939i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(131,⋅)\chi_{220} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.343+0.939i)(2,\ 220,\ (\ :1/2),\ 0.343 + 0.939i)

Particular Values

L(1)L(1) ≈\approx 0.796331−0.556926i0.796331 - 0.556926i
L(12)L(\frac12) ≈\approx 0.796331−0.556926i0.796331 - 0.556926i
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+(0.960+1.03i)T 1 + (0.960 + 1.03i)T
5 1−T 1 - T
11 1+(−3.01+1.37i)T 1 + (-3.01 + 1.37i)T
good3 1+0.450iT−3T2 1 + 0.450iT - 3T^{2}
7 1+1.25T+7T2 1 + 1.25T + 7T^{2}
13 1+3.65iT−13T2 1 + 3.65iT - 13T^{2}
17 1−1.35iT−17T2 1 - 1.35iT - 17T^{2}
19 1−1.25T+19T2 1 - 1.25T + 19T^{2}
23 1+3.06iT−23T2 1 + 3.06iT - 23T^{2}
29 1−2.29iT−29T2 1 - 2.29iT - 29T^{2}
31 1+5.36iT−31T2 1 + 5.36iT - 31T^{2}
37 1+3.17T+37T2 1 + 3.17T + 37T^{2}
41 1−11.1iT−41T2 1 - 11.1iT - 41T^{2}
43 1−4.16T+43T2 1 - 4.16T + 43T^{2}
47 1−9.18iT−47T2 1 - 9.18iT - 47T^{2}
53 1−4.41T+53T2 1 - 4.41T + 53T^{2}
59 1−8.87iT−59T2 1 - 8.87iT - 59T^{2}
61 1−5.01iT−61T2 1 - 5.01iT - 61T^{2}
67 1−9.46iT−67T2 1 - 9.46iT - 67T^{2}
71 1+10.5iT−71T2 1 + 10.5iT - 71T^{2}
73 1+10.3iT−73T2 1 + 10.3iT - 73T^{2}
79 1+12.9T+79T2 1 + 12.9T + 79T^{2}
83 1+6.68T+83T2 1 + 6.68T + 83T^{2}
89 1+9.39T+89T2 1 + 9.39T + 89T^{2}
97 1+18.4T+97T2 1 + 18.4T + 97T^{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

−12.14040112385413646575903457759, −10.99263820114980407191616391263, −10.08925262041510538407858183099, −9.386297445177562480619476470007, −8.287971834615081924493340883407, −7.20304791236222371755833824330, −6.11440847365051470276656342851, −4.27632374393016544779193315144, −2.94324033466690018803420983697, −1.25823738211269500436308085280, 1.69148623158413078306913117523, 4.06603420002457643688347955046, 5.33798880440075497666870362337, 6.69731861199770789060533113746, 7.16427773524480017182779222821, 8.750811927904643496487863185441, 9.552868509545512698419348275934, 10.07676086243640397745089178157, 11.30641115083137010437546634956, 12.47840139716684268869231767815

Graph of the ZZ-function along the critical line