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Properties

Label 2-220-44.7-c1-0-8
Degree 22
Conductor 220220
Sign −0.299−0.954i-0.299 - 0.954i
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  + (0.432 + 1.34i)2-s + (2.09 + 0.680i)3-s + (−1.62 + 1.16i)4-s + (−0.809 + 0.587i)5-s + (−0.0101 + 3.11i)6-s + (0.548 + 1.68i)7-s + (−2.27 − 1.68i)8-s + (1.49 + 1.08i)9-s + (−1.14 − 0.834i)10-s + (−0.218 − 3.30i)11-s + (−4.19 + 1.33i)12-s + (0.368 − 0.506i)13-s + (−2.03 + 1.46i)14-s + (−2.09 + 0.680i)15-s + (1.28 − 3.78i)16-s + (4.68 + 6.44i)17-s + ⋯
L(s)  = 1  + (0.305 + 0.952i)2-s + (1.20 + 0.392i)3-s + (−0.812 + 0.582i)4-s + (−0.361 + 0.262i)5-s + (−0.00413 + 1.27i)6-s + (0.207 + 0.638i)7-s + (−0.803 − 0.595i)8-s + (0.499 + 0.362i)9-s + (−0.360 − 0.264i)10-s + (−0.0659 − 0.997i)11-s + (−1.21 + 0.385i)12-s + (0.102 − 0.140i)13-s + (−0.544 + 0.392i)14-s + (−0.540 + 0.175i)15-s + (0.321 − 0.946i)16-s + (1.13 + 1.56i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.299−0.954i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((−0.299−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.299−0.954i-0.299 - 0.954i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(51,⋅)\chi_{220} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), −0.299−0.954i)(2,\ 220,\ (\ :1/2),\ -0.299 - 0.954i)

Particular Values

L(1)L(1) ≈\approx 1.03471+1.40923i1.03471 + 1.40923i
L(12)L(\frac12) ≈\approx 1.03471+1.40923i1.03471 + 1.40923i
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+(−0.432−1.34i)T 1 + (-0.432 - 1.34i)T
5 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1+(0.218+3.30i)T 1 + (0.218 + 3.30i)T
good3 1+(−2.09−0.680i)T+(2.42+1.76i)T2 1 + (-2.09 - 0.680i)T + (2.42 + 1.76i)T^{2}
7 1+(−0.548−1.68i)T+(−5.66+4.11i)T2 1 + (-0.548 - 1.68i)T + (-5.66 + 4.11i)T^{2}
13 1+(−0.368+0.506i)T+(−4.01−12.3i)T2 1 + (-0.368 + 0.506i)T + (-4.01 - 12.3i)T^{2}
17 1+(−4.68−6.44i)T+(−5.25+16.1i)T2 1 + (-4.68 - 6.44i)T + (-5.25 + 16.1i)T^{2}
19 1+(−1.50+4.62i)T+(−15.3−11.1i)T2 1 + (-1.50 + 4.62i)T + (-15.3 - 11.1i)T^{2}
23 1+4.01iT−23T2 1 + 4.01iT - 23T^{2}
29 1+(0.923−0.300i)T+(23.4−17.0i)T2 1 + (0.923 - 0.300i)T + (23.4 - 17.0i)T^{2}
31 1+(2.21−3.05i)T+(−9.57−29.4i)T2 1 + (2.21 - 3.05i)T + (-9.57 - 29.4i)T^{2}
37 1+(−1.35−4.18i)T+(−29.9+21.7i)T2 1 + (-1.35 - 4.18i)T + (-29.9 + 21.7i)T^{2}
41 1+(7.91+2.57i)T+(33.1+24.0i)T2 1 + (7.91 + 2.57i)T + (33.1 + 24.0i)T^{2}
43 1−8.42T+43T2 1 - 8.42T + 43T^{2}
47 1+(7.43+2.41i)T+(38.0+27.6i)T2 1 + (7.43 + 2.41i)T + (38.0 + 27.6i)T^{2}
53 1+(7.51+5.45i)T+(16.3+50.4i)T2 1 + (7.51 + 5.45i)T + (16.3 + 50.4i)T^{2}
59 1+(2.45−0.798i)T+(47.7−34.6i)T2 1 + (2.45 - 0.798i)T + (47.7 - 34.6i)T^{2}
61 1+(−5.77−7.95i)T+(−18.8+58.0i)T2 1 + (-5.77 - 7.95i)T + (-18.8 + 58.0i)T^{2}
67 1+7.43iT−67T2 1 + 7.43iT - 67T^{2}
71 1+(1.04+1.44i)T+(−21.9+67.5i)T2 1 + (1.04 + 1.44i)T + (-21.9 + 67.5i)T^{2}
73 1+(3.83−1.24i)T+(59.0−42.9i)T2 1 + (3.83 - 1.24i)T + (59.0 - 42.9i)T^{2}
79 1+(−2.58−1.87i)T+(24.4+75.1i)T2 1 + (-2.58 - 1.87i)T + (24.4 + 75.1i)T^{2}
83 1+(−6.14+4.46i)T+(25.6−78.9i)T2 1 + (-6.14 + 4.46i)T + (25.6 - 78.9i)T^{2}
89 1+6.28T+89T2 1 + 6.28T + 89T^{2}
97 1+(−4.44−3.23i)T+(29.9+92.2i)T2 1 + (-4.44 - 3.23i)T + (29.9 + 92.2i)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

−12.89117515523684017696166346025, −11.81410661661471810520689903636, −10.43070376746651034797880349017, −9.167937270078684718505649841779, −8.440818045531642945773951962863, −7.901773788496898132070206740206, −6.46389027198463492098946901758, −5.31330648831270028557240673714, −3.82589693985355234562729882435, −2.98034603420950913926048711534, 1.55274375126064292882190770778, 3.01524094365817335754302515676, 4.07371838323940531584049554323, 5.34912237688308720129445841938, 7.37668480590789113062344254689, 7.996366551439850715092230136532, 9.339235515811333709269799791263, 9.843640272701536646224709124777, 11.19904516796628647829935799774, 12.13240812128749095083176541770

Graph of the ZZ-function along the critical line