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Properties

Label 2-220-55.39-c2-0-0
Degree 22
Conductor 220220
Sign −0.398+0.917i-0.398 + 0.917i
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  + (−2.63 + 3.62i)3-s + (−3.75 + 3.29i)5-s + (−5.27 + 3.83i)7-s + (−3.42 − 10.5i)9-s + (8.17 − 7.36i)11-s + (−1.48 − 4.55i)13-s + (−2.04 − 22.3i)15-s + (2.76 − 8.50i)17-s + (−2.05 + 2.83i)19-s − 29.2i·21-s + 29.8i·23-s + (3.27 − 24.7i)25-s + (8.89 + 2.89i)27-s + (−30.7 − 42.3i)29-s + (2.88 + 8.88i)31-s + ⋯
L(s)  = 1  + (−0.878 + 1.20i)3-s + (−0.751 + 0.659i)5-s + (−0.754 + 0.548i)7-s + (−0.380 − 1.17i)9-s + (0.742 − 0.669i)11-s + (−0.113 − 0.350i)13-s + (−0.136 − 1.48i)15-s + (0.162 − 0.500i)17-s + (−0.108 + 0.149i)19-s − 1.39i·21-s + 1.29i·23-s + (0.130 − 0.991i)25-s + (0.329 + 0.107i)27-s + (−1.06 − 1.46i)29-s + (0.0931 + 0.286i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.398+0.917i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1) L(s)=((−0.398+0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.398+0.917i-0.398 + 0.917i
Analytic conductor: 5.994565.99456
Root analytic conductor: 2.448382.44838
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ220(149,⋅)\chi_{220} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1), −0.398+0.917i)(2,\ 220,\ (\ :1),\ -0.398 + 0.917i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0651339−0.0993079i0.0651339 - 0.0993079i
L(12)L(\frac12) ≈\approx 0.0651339−0.0993079i0.0651339 - 0.0993079i
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+(3.75−3.29i)T 1 + (3.75 - 3.29i)T
11 1+(−8.17+7.36i)T 1 + (-8.17 + 7.36i)T
good3 1+(2.63−3.62i)T+(−2.78−8.55i)T2 1 + (2.63 - 3.62i)T + (-2.78 - 8.55i)T^{2}
7 1+(5.27−3.83i)T+(15.1−46.6i)T2 1 + (5.27 - 3.83i)T + (15.1 - 46.6i)T^{2}
13 1+(1.48+4.55i)T+(−136.+99.3i)T2 1 + (1.48 + 4.55i)T + (-136. + 99.3i)T^{2}
17 1+(−2.76+8.50i)T+(−233.−169.i)T2 1 + (-2.76 + 8.50i)T + (-233. - 169. i)T^{2}
19 1+(2.05−2.83i)T+(−111.−343.i)T2 1 + (2.05 - 2.83i)T + (-111. - 343. i)T^{2}
23 1−29.8iT−529T2 1 - 29.8iT - 529T^{2}
29 1+(30.7+42.3i)T+(−259.+799.i)T2 1 + (30.7 + 42.3i)T + (-259. + 799. i)T^{2}
31 1+(−2.88−8.88i)T+(−777.+564.i)T2 1 + (-2.88 - 8.88i)T + (-777. + 564. i)T^{2}
37 1+(34.2+47.0i)T+(−423.+1.30e3i)T2 1 + (34.2 + 47.0i)T + (-423. + 1.30e3i)T^{2}
41 1+(33.0−45.5i)T+(−519.−1.59e3i)T2 1 + (33.0 - 45.5i)T + (-519. - 1.59e3i)T^{2}
43 1−26.6T+1.84e3T2 1 - 26.6T + 1.84e3T^{2}
47 1+(0.426−0.586i)T+(−682.−2.10e3i)T2 1 + (0.426 - 0.586i)T + (-682. - 2.10e3i)T^{2}
53 1+(67.9−22.0i)T+(2.27e3−1.65e3i)T2 1 + (67.9 - 22.0i)T + (2.27e3 - 1.65e3i)T^{2}
59 1+(81.8−59.4i)T+(1.07e3−3.31e3i)T2 1 + (81.8 - 59.4i)T + (1.07e3 - 3.31e3i)T^{2}
61 1+(−68.3−22.2i)T+(3.01e3+2.18e3i)T2 1 + (-68.3 - 22.2i)T + (3.01e3 + 2.18e3i)T^{2}
67 1+7.36iT−4.48e3T2 1 + 7.36iT - 4.48e3T^{2}
71 1+(4.14−12.7i)T+(−4.07e3−2.96e3i)T2 1 + (4.14 - 12.7i)T + (-4.07e3 - 2.96e3i)T^{2}
73 1+(−76.1+55.3i)T+(1.64e3−5.06e3i)T2 1 + (-76.1 + 55.3i)T + (1.64e3 - 5.06e3i)T^{2}
79 1+(48.3−15.7i)T+(5.04e3−3.66e3i)T2 1 + (48.3 - 15.7i)T + (5.04e3 - 3.66e3i)T^{2}
83 1+(−27.8+85.8i)T+(−5.57e3−4.04e3i)T2 1 + (-27.8 + 85.8i)T + (-5.57e3 - 4.04e3i)T^{2}
89 1−12.2T+7.92e3T2 1 - 12.2T + 7.92e3T^{2}
97 1+(67.3−21.8i)T+(7.61e3−5.53e3i)T2 1 + (67.3 - 21.8i)T + (7.61e3 - 5.53e3i)T^{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

−12.30454677205199348173474277327, −11.52947779199769154346570479117, −10.93374360860276050700367401253, −9.866582678015638531042711954637, −9.156457909319688093676591824782, −7.67586617841312076867140183550, −6.33279445143760221735441927246, −5.51096984036792827752105881437, −4.08388716531155750342110666135, −3.19447034995246379949126560168, 0.07494314699948993333617218527, 1.51871619462684986599901752832, 3.79566039590959682416501385374, 5.10783148960583270343205028501, 6.59465843439558850586739512136, 7.01126396957020550283804735214, 8.188679204299592400817528548468, 9.374469114325219599960009212437, 10.69641889747348596044698454397, 11.67372803736072838016760660951

Graph of the ZZ-function along the critical line