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Properties

Label 2-220-220.219-c1-0-9
Degree 22
Conductor 220220
Sign 0.205−0.978i0.205 - 0.978i
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  + (−1.33 + 0.478i)2-s + 1.30·3-s + (1.54 − 1.27i)4-s + (−0.393 + 2.20i)5-s + (−1.73 + 0.624i)6-s + 1.57i·7-s + (−1.44 + 2.43i)8-s − 1.29·9-s + (−0.529 − 3.11i)10-s + (2.49 + 2.18i)11-s + (2.01 − 1.66i)12-s + 1.50·13-s + (−0.755 − 2.09i)14-s + (−0.513 + 2.87i)15-s + (0.755 − 3.92i)16-s − 4.38·17-s + ⋯
L(s)  = 1  + (−0.941 + 0.338i)2-s + 0.753·3-s + (0.770 − 0.636i)4-s + (−0.175 + 0.984i)5-s + (−0.708 + 0.254i)6-s + 0.596i·7-s + (−0.509 + 0.860i)8-s − 0.432·9-s + (−0.167 − 0.985i)10-s + (0.751 + 0.659i)11-s + (0.580 − 0.479i)12-s + 0.416·13-s + (−0.201 − 0.561i)14-s + (−0.132 + 0.741i)15-s + (0.188 − 0.982i)16-s − 1.06·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.205−0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.205−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.205−0.978i0.205 - 0.978i
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), 0.205−0.978i)(2,\ 220,\ (\ :1/2),\ 0.205 - 0.978i)

Particular Values

L(1)L(1) ≈\approx 0.751698+0.610359i0.751698 + 0.610359i
L(12)L(\frac12) ≈\approx 0.751698+0.610359i0.751698 + 0.610359i
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+(1.33−0.478i)T 1 + (1.33 - 0.478i)T
5 1+(0.393−2.20i)T 1 + (0.393 - 2.20i)T
11 1+(−2.49−2.18i)T 1 + (-2.49 - 2.18i)T
good3 1−1.30T+3T2 1 - 1.30T + 3T^{2}
7 1−1.57iT−7T2 1 - 1.57iT - 7T^{2}
13 1−1.50T+13T2 1 - 1.50T + 13T^{2}
17 1+4.38T+17T2 1 + 4.38T + 17T^{2}
19 1−5.72T+19T2 1 - 5.72T + 19T^{2}
23 1−5.32T+23T2 1 - 5.32T + 23T^{2}
29 1−6.34iT−29T2 1 - 6.34iT - 29T^{2}
31 1−4.48iT−31T2 1 - 4.48iT - 31T^{2}
37 1+6.78iT−37T2 1 + 6.78iT - 37T^{2}
41 1+8.40iT−41T2 1 + 8.40iT - 41T^{2}
43 1+11.5iT−43T2 1 + 11.5iT - 43T^{2}
47 1−4.30T+47T2 1 - 4.30T + 47T^{2}
53 1−2.38iT−53T2 1 - 2.38iT - 53T^{2}
59 1+7.74iT−59T2 1 + 7.74iT - 59T^{2}
61 1+4.55iT−61T2 1 + 4.55iT - 61T^{2}
67 1+5.32T+67T2 1 + 5.32T + 67T^{2}
71 1+6.85iT−71T2 1 + 6.85iT - 71T^{2}
73 1+2.63T+73T2 1 + 2.63T + 73T^{2}
79 1+4.98T+79T2 1 + 4.98T + 79T^{2}
83 1−1.78iT−83T2 1 - 1.78iT - 83T^{2}
89 1+2.72T+89T2 1 + 2.72T + 89T^{2}
97 1−4.40iT−97T2 1 - 4.40iT - 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

−12.19778735797324866299201688756, −11.29073370875188108663350789451, −10.48998958704110463185426712149, −9.102366899684564633385417326866, −8.890647366156406703842207693136, −7.45486686862180651099447096056, −6.80431005865181383801167615863, −5.51034168669342553732008438524, −3.36602818591643755767482445815, −2.16166842567365339613738215129, 1.10464868421829108742534374899, 2.97490051431489866522112684772, 4.21367371859033623376301866996, 6.09153184365118856609519761181, 7.48153936028077550895555166582, 8.382782450515358549892857440964, 9.037095386002341257720084642386, 9.785407007206793221669777938478, 11.32424079482241075067570187389, 11.63834132137604500236834667155

Graph of the ZZ-function along the critical line