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Properties

Label 2-220-20.3-c1-0-19
Degree 22
Conductor 220220
Sign 0.865+0.500i0.865 + 0.500i
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  + (1.31 − 0.513i)2-s + (−0.912 + 0.912i)3-s + (1.47 − 1.35i)4-s + (1.46 − 1.68i)5-s + (−0.733 + 1.67i)6-s + (−0.0653 − 0.0653i)7-s + (1.24 − 2.54i)8-s + 1.33i·9-s + (1.06 − 2.97i)10-s − i·11-s + (−0.107 + 2.57i)12-s + (0.473 + 0.473i)13-s + (−0.119 − 0.0524i)14-s + (0.204 + 2.87i)15-s + (0.332 − 3.98i)16-s + (−1.01 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.931 − 0.363i)2-s + (−0.526 + 0.526i)3-s + (0.735 − 0.677i)4-s + (0.655 − 0.755i)5-s + (−0.299 + 0.682i)6-s + (−0.0246 − 0.0246i)7-s + (0.439 − 0.898i)8-s + 0.444i·9-s + (0.335 − 0.941i)10-s − 0.301i·11-s + (−0.0309 + 0.744i)12-s + (0.131 + 0.131i)13-s + (−0.0319 − 0.0140i)14-s + (0.0528 + 0.743i)15-s + (0.0830 − 0.996i)16-s + (−0.245 + 0.245i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.865+0.500i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.865+0.500i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.865+0.500i0.865 + 0.500i
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), 0.865+0.500i)(2,\ 220,\ (\ :1/2),\ 0.865 + 0.500i)

Particular Values

L(1)L(1) ≈\approx 1.83354−0.491808i1.83354 - 0.491808i
L(12)L(\frac12) ≈\approx 1.83354−0.491808i1.83354 - 0.491808i
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+(−1.31+0.513i)T 1 + (-1.31 + 0.513i)T
5 1+(−1.46+1.68i)T 1 + (-1.46 + 1.68i)T
11 1+iT 1 + iT
good3 1+(0.912−0.912i)T−3iT2 1 + (0.912 - 0.912i)T - 3iT^{2}
7 1+(0.0653+0.0653i)T+7iT2 1 + (0.0653 + 0.0653i)T + 7iT^{2}
13 1+(−0.473−0.473i)T+13iT2 1 + (-0.473 - 0.473i)T + 13iT^{2}
17 1+(1.01−1.01i)T−17iT2 1 + (1.01 - 1.01i)T - 17iT^{2}
19 1−2.65T+19T2 1 - 2.65T + 19T^{2}
23 1+(5.32−5.32i)T−23iT2 1 + (5.32 - 5.32i)T - 23iT^{2}
29 1−5.94iT−29T2 1 - 5.94iT - 29T^{2}
31 1−1.44iT−31T2 1 - 1.44iT - 31T^{2}
37 1+(5.27−5.27i)T−37iT2 1 + (5.27 - 5.27i)T - 37iT^{2}
41 1+5.13T+41T2 1 + 5.13T + 41T^{2}
43 1+(4.06−4.06i)T−43iT2 1 + (4.06 - 4.06i)T - 43iT^{2}
47 1+(7.89+7.89i)T+47iT2 1 + (7.89 + 7.89i)T + 47iT^{2}
53 1+(−2.60−2.60i)T+53iT2 1 + (-2.60 - 2.60i)T + 53iT^{2}
59 1−7.70T+59T2 1 - 7.70T + 59T^{2}
61 1−7.06T+61T2 1 - 7.06T + 61T^{2}
67 1+(−5.47−5.47i)T+67iT2 1 + (-5.47 - 5.47i)T + 67iT^{2}
71 1+1.80iT−71T2 1 + 1.80iT - 71T^{2}
73 1+(−5.16−5.16i)T+73iT2 1 + (-5.16 - 5.16i)T + 73iT^{2}
79 1−14.2T+79T2 1 - 14.2T + 79T^{2}
83 1+(−11.9+11.9i)T−83iT2 1 + (-11.9 + 11.9i)T - 83iT^{2}
89 1+5.69iT−89T2 1 + 5.69iT - 89T^{2}
97 1+(0.398−0.398i)T−97iT2 1 + (0.398 - 0.398i)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

−12.11958497397502665443461383386, −11.42543456784648680343337957978, −10.33819835365175548490084088577, −9.723847458896414653144354802389, −8.303915177972996513214522828321, −6.70404330228481278574375419233, −5.49962230331611239823887521395, −5.00109772773873541451647611080, −3.65296834201862172864947065186, −1.79140148749787274270072734663, 2.22978421346614828705390282665, 3.70621409831943103843164909669, 5.28531815795923366984717247156, 6.29121920684143919758051267333, 6.84482919533330499286492839889, 7.989354990795769072340058121419, 9.572814531805546894399298231278, 10.72596727433304238946470118086, 11.68690842864470167210631757891, 12.40726944964209372888768418275

Graph of the ZZ-function along the critical line