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Properties

Label 2-220-220.219-c1-0-12
Degree 22
Conductor 220220
Sign 0.997+0.0750i0.997 + 0.0750i
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  + (−0.896 − 1.09i)2-s + 2.84·3-s + (−0.393 + 1.96i)4-s + (−1.64 + 1.51i)5-s + (−2.54 − 3.11i)6-s + 3.37i·7-s + (2.49 − 1.32i)8-s + 5.08·9-s + (3.13 + 0.449i)10-s + (1.92 − 2.70i)11-s + (−1.11 + 5.57i)12-s + 2.23·13-s + (3.69 − 3.02i)14-s + (−4.68 + 4.29i)15-s + (−3.69 − 1.54i)16-s + 2.76·17-s + ⋯
L(s)  = 1  + (−0.633 − 0.773i)2-s + 1.64·3-s + (−0.196 + 0.980i)4-s + (−0.737 + 0.675i)5-s + (−1.04 − 1.26i)6-s + 1.27i·7-s + (0.883 − 0.469i)8-s + 1.69·9-s + (0.989 + 0.142i)10-s + (0.580 − 0.814i)11-s + (−0.322 + 1.60i)12-s + 0.618·13-s + (0.986 − 0.808i)14-s + (−1.21 + 1.10i)15-s + (−0.922 − 0.385i)16-s + 0.670·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.997+0.0750i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0750i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.997+0.0750i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0750i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.997+0.0750i0.997 + 0.0750i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(219,⋅)\chi_{220} (219, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.997+0.0750i)(2,\ 220,\ (\ :1/2),\ 0.997 + 0.0750i)

Particular Values

L(1)L(1) ≈\approx 1.34063−0.0504011i1.34063 - 0.0504011i
L(12)L(\frac12) ≈\approx 1.34063−0.0504011i1.34063 - 0.0504011i
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+(0.896+1.09i)T 1 + (0.896 + 1.09i)T
5 1+(1.64−1.51i)T 1 + (1.64 - 1.51i)T
11 1+(−1.92+2.70i)T 1 + (-1.92 + 2.70i)T
good3 1−2.84T+3T2 1 - 2.84T + 3T^{2}
7 1−3.37iT−7T2 1 - 3.37iT - 7T^{2}
13 1−2.23T+13T2 1 - 2.23T + 13T^{2}
17 1−2.76T+17T2 1 - 2.76T + 17T^{2}
19 1+7.85T+19T2 1 + 7.85T + 19T^{2}
23 1−0.606T+23T2 1 - 0.606T + 23T^{2}
29 1+7.54iT−29T2 1 + 7.54iT - 29T^{2}
31 1−2.02iT−31T2 1 - 2.02iT - 31T^{2}
37 1−1.18iT−37T2 1 - 1.18iT - 37T^{2}
41 1+2.04iT−41T2 1 + 2.04iT - 41T^{2}
43 1+3.09iT−43T2 1 + 3.09iT - 43T^{2}
47 1+8.76T+47T2 1 + 8.76T + 47T^{2}
53 1+4.21iT−53T2 1 + 4.21iT - 53T^{2}
59 1−4.33iT−59T2 1 - 4.33iT - 59T^{2}
61 1−2.84iT−61T2 1 - 2.84iT - 61T^{2}
67 1+0.606T+67T2 1 + 0.606T + 67T^{2}
71 1+9.55iT−71T2 1 + 9.55iT - 71T^{2}
73 1−6.00T+73T2 1 - 6.00T + 73T^{2}
79 1+3.84T+79T2 1 + 3.84T + 79T^{2}
83 1−8.39iT−83T2 1 - 8.39iT - 83T^{2}
89 1−8.67T+89T2 1 - 8.67T + 89T^{2}
97 1−3.02iT−97T2 1 - 3.02iT - 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

−12.16059329638475988143896140354, −11.29466666962254840877195128375, −10.21104656800196933954365353555, −9.060980603681854522765763242184, −8.490249722086344046164095965007, −7.902569112762622516797644067631, −6.46492314923051080556676551242, −4.02762784608909246488957027160, −3.17331313025030955062920568372, −2.17865559169244424281991082495, 1.49039989090425321262231695610, 3.76556554282840678620146706888, 4.58692671501313617649581694727, 6.72025811013808639620047047820, 7.57680155006708483643416221867, 8.300777216681132939461984435943, 9.037422790840353500563982813780, 9.961443574900448728051167479139, 10.97223650459976155203458717544, 12.70564078736766401037377490047

Graph of the ZZ-function along the critical line