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Properties

Label 2-220-20.3-c1-0-2
Degree 22
Conductor 220220
Sign 0.799−0.600i0.799 - 0.600i
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  + (−0.354 − 1.36i)2-s + (−1.16 + 1.16i)3-s + (−1.74 + 0.971i)4-s + (−1.42 − 1.72i)5-s + (2.00 + 1.18i)6-s + (1.60 + 1.60i)7-s + (1.94 + 2.04i)8-s + 0.282i·9-s + (−1.85 + 2.56i)10-s + i·11-s + (0.906 − 3.17i)12-s + (3.94 + 3.94i)13-s + (1.62 − 2.76i)14-s + (3.66 + 0.350i)15-s + (2.11 − 3.39i)16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (−0.250 − 0.968i)2-s + (−0.673 + 0.673i)3-s + (−0.874 + 0.485i)4-s + (−0.636 − 0.771i)5-s + (0.820 + 0.482i)6-s + (0.605 + 0.605i)7-s + (0.689 + 0.724i)8-s + 0.0940i·9-s + (−0.586 + 0.809i)10-s + 0.301i·11-s + (0.261 − 0.915i)12-s + (1.09 + 1.09i)13-s + (0.434 − 0.737i)14-s + (0.947 + 0.0903i)15-s + (0.528 − 0.848i)16-s + (−1.17 + 1.17i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.799−0.600i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.799−0.600i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.799−0.600i0.799 - 0.600i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(23,⋅)\chi_{220} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.799−0.600i)(2,\ 220,\ (\ :1/2),\ 0.799 - 0.600i)

Particular Values

L(1)L(1) ≈\approx 0.637662+0.212639i0.637662 + 0.212639i
L(12)L(\frac12) ≈\approx 0.637662+0.212639i0.637662 + 0.212639i
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+(0.354+1.36i)T 1 + (0.354 + 1.36i)T
5 1+(1.42+1.72i)T 1 + (1.42 + 1.72i)T
11 1−iT 1 - iT
good3 1+(1.16−1.16i)T−3iT2 1 + (1.16 - 1.16i)T - 3iT^{2}
7 1+(−1.60−1.60i)T+7iT2 1 + (-1.60 - 1.60i)T + 7iT^{2}
13 1+(−3.94−3.94i)T+13iT2 1 + (-3.94 - 3.94i)T + 13iT^{2}
17 1+(4.86−4.86i)T−17iT2 1 + (4.86 - 4.86i)T - 17iT^{2}
19 1−5.54T+19T2 1 - 5.54T + 19T^{2}
23 1+(3.01−3.01i)T−23iT2 1 + (3.01 - 3.01i)T - 23iT^{2}
29 1+6.65iT−29T2 1 + 6.65iT - 29T^{2}
31 1−0.619iT−31T2 1 - 0.619iT - 31T^{2}
37 1+(−0.294+0.294i)T−37iT2 1 + (-0.294 + 0.294i)T - 37iT^{2}
41 1+2.27T+41T2 1 + 2.27T + 41T^{2}
43 1+(2.52−2.52i)T−43iT2 1 + (2.52 - 2.52i)T - 43iT^{2}
47 1+(−8.42−8.42i)T+47iT2 1 + (-8.42 - 8.42i)T + 47iT^{2}
53 1+(−2.22−2.22i)T+53iT2 1 + (-2.22 - 2.22i)T + 53iT^{2}
59 1+1.02T+59T2 1 + 1.02T + 59T^{2}
61 1+4.52T+61T2 1 + 4.52T + 61T^{2}
67 1+(3.45+3.45i)T+67iT2 1 + (3.45 + 3.45i)T + 67iT^{2}
71 1−11.6iT−71T2 1 - 11.6iT - 71T^{2}
73 1+(−3.57−3.57i)T+73iT2 1 + (-3.57 - 3.57i)T + 73iT^{2}
79 1−2.91T+79T2 1 - 2.91T + 79T^{2}
83 1+(−9.90+9.90i)T−83iT2 1 + (-9.90 + 9.90i)T - 83iT^{2}
89 1+10.6iT−89T2 1 + 10.6iT - 89T^{2}
97 1+(0.328−0.328i)T−97iT2 1 + (0.328 - 0.328i)T - 97iT^{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

−11.82509622980105370575464985428, −11.56612355178822024037591133195, −10.71474422176355619871836919019, −9.512860809632527566546148785680, −8.696993599214977583794867206143, −7.81737852667245224351661456473, −5.78314296886247503140283202954, −4.62614042031153995175681163074, −3.96275256325090333761856762225, −1.76171195040172244662055396504, 0.72430699420033468704896578392, 3.62492579445557852930175202565, 5.13005292666990074950801979224, 6.30005004215301743199899314779, 7.11698473101565287176271336495, 7.83369952270886482603732917710, 8.937389007123087755887957409451, 10.45280750523032070525086321214, 11.12737794653577316907660306753, 12.11786042130547443126178139645

Graph of the ZZ-function along the critical line