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Properties

Label 2-220-20.3-c1-0-17
Degree 22
Conductor 220220
Sign 0.375+0.926i0.375 + 0.926i
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  + (−0.171 + 1.40i)2-s + (−1.44 + 1.44i)3-s + (−1.94 − 0.481i)4-s + (0.849 − 2.06i)5-s + (−1.77 − 2.27i)6-s + (−3.60 − 3.60i)7-s + (1.00 − 2.64i)8-s − 1.16i·9-s + (2.75 + 1.54i)10-s + i·11-s + (3.49 − 2.10i)12-s + (−2.27 − 2.27i)13-s + (5.67 − 4.44i)14-s + (1.76 + 4.21i)15-s + (3.53 + 1.86i)16-s + (−3.21 + 3.21i)17-s + ⋯
L(s)  = 1  + (−0.121 + 0.992i)2-s + (−0.833 + 0.833i)3-s + (−0.970 − 0.240i)4-s + (0.379 − 0.925i)5-s + (−0.726 − 0.928i)6-s + (−1.36 − 1.36i)7-s + (0.356 − 0.934i)8-s − 0.388i·9-s + (0.872 + 0.489i)10-s + 0.301i·11-s + (1.00 − 0.608i)12-s + (−0.630 − 0.630i)13-s + (1.51 − 1.18i)14-s + (0.454 + 1.08i)15-s + (0.884 + 0.467i)16-s + (−0.780 + 0.780i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.375+0.926i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.375+0.926i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.375+0.926i0.375 + 0.926i
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), 0.375+0.926i)(2,\ 220,\ (\ :1/2),\ 0.375 + 0.926i)

Particular Values

L(1)L(1) ≈\approx 0.242530−0.163489i0.242530 - 0.163489i
L(12)L(\frac12) ≈\approx 0.242530−0.163489i0.242530 - 0.163489i
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+(0.171−1.40i)T 1 + (0.171 - 1.40i)T
5 1+(−0.849+2.06i)T 1 + (-0.849 + 2.06i)T
11 1−iT 1 - iT
good3 1+(1.44−1.44i)T−3iT2 1 + (1.44 - 1.44i)T - 3iT^{2}
7 1+(3.60+3.60i)T+7iT2 1 + (3.60 + 3.60i)T + 7iT^{2}
13 1+(2.27+2.27i)T+13iT2 1 + (2.27 + 2.27i)T + 13iT^{2}
17 1+(3.21−3.21i)T−17iT2 1 + (3.21 - 3.21i)T - 17iT^{2}
19 1+1.17T+19T2 1 + 1.17T + 19T^{2}
23 1+(−1.72+1.72i)T−23iT2 1 + (-1.72 + 1.72i)T - 23iT^{2}
29 1+8.70iT−29T2 1 + 8.70iT - 29T^{2}
31 1−2.23iT−31T2 1 - 2.23iT - 31T^{2}
37 1+(4.21−4.21i)T−37iT2 1 + (4.21 - 4.21i)T - 37iT^{2}
41 1+1.59T+41T2 1 + 1.59T + 41T^{2}
43 1+(−4.09+4.09i)T−43iT2 1 + (-4.09 + 4.09i)T - 43iT^{2}
47 1+(−0.691−0.691i)T+47iT2 1 + (-0.691 - 0.691i)T + 47iT^{2}
53 1+(3.07+3.07i)T+53iT2 1 + (3.07 + 3.07i)T + 53iT^{2}
59 1+3.06T+59T2 1 + 3.06T + 59T^{2}
61 1+3.44T+61T2 1 + 3.44T + 61T^{2}
67 1+(−4.51−4.51i)T+67iT2 1 + (-4.51 - 4.51i)T + 67iT^{2}
71 1+10.0iT−71T2 1 + 10.0iT - 71T^{2}
73 1+(−6.84−6.84i)T+73iT2 1 + (-6.84 - 6.84i)T + 73iT^{2}
79 1−0.973T+79T2 1 - 0.973T + 79T^{2}
83 1+(0.453−0.453i)T−83iT2 1 + (0.453 - 0.453i)T - 83iT^{2}
89 1−4.23iT−89T2 1 - 4.23iT - 89T^{2}
97 1+(−7.14+7.14i)T−97iT2 1 + (-7.14 + 7.14i)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

−12.42557861063635009225928170288, −10.63150230332806463225468642174, −10.05655231877962870052649822892, −9.380573912360657024900208697317, −8.052438874087365811944688331065, −6.79171890752715109673987250764, −5.90732999239805729153565720240, −4.76114498163230636372873513441, −4.00623259357032257410502083974, −0.26953304831798036232297033671, 2.19895692645544916761152041771, 3.24728603012432998188614985357, 5.32622021161420487200850592082, 6.32311533345547062408848724748, 7.15935407681511123757414294227, 9.029295311562201642243854706141, 9.536435546308381189733312768068, 10.80581726178955947089095483730, 11.60677977771844290876122952362, 12.36574764909709453128650355821

Graph of the ZZ-function along the critical line