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Properties

Label 2-220-44.43-c1-0-20
Degree 22
Conductor 220220
Sign 0.260+0.965i0.260 + 0.965i
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  + (1.39 − 0.249i)2-s − 2.27i·3-s + (1.87 − 0.695i)4-s + 5-s + (−0.567 − 3.16i)6-s − 3.82·7-s + (2.43 − 1.43i)8-s − 2.15·9-s + (1.39 − 0.249i)10-s + (−0.302 + 3.30i)11-s + (−1.57 − 4.25i)12-s + 4.63i·13-s + (−5.32 + 0.956i)14-s − 2.27i·15-s + (3.03 − 2.60i)16-s − 0.686i·17-s + ⋯
L(s)  = 1  + (0.984 − 0.176i)2-s − 1.31i·3-s + (0.937 − 0.347i)4-s + 0.447·5-s + (−0.231 − 1.29i)6-s − 1.44·7-s + (0.861 − 0.507i)8-s − 0.719·9-s + (0.440 − 0.0789i)10-s + (−0.0911 + 0.995i)11-s + (−0.455 − 1.22i)12-s + 1.28i·13-s + (−1.42 + 0.255i)14-s − 0.586i·15-s + (0.758 − 0.651i)16-s − 0.166i·17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.260+0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.260+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.260+0.965i0.260 + 0.965i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(131,⋅)\chi_{220} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.260+0.965i)(2,\ 220,\ (\ :1/2),\ 0.260 + 0.965i)

Particular Values

L(1)L(1) ≈\approx 1.63384−1.25120i1.63384 - 1.25120i
L(12)L(\frac12) ≈\approx 1.63384−1.25120i1.63384 - 1.25120i
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+(−1.39+0.249i)T 1 + (-1.39 + 0.249i)T
5 1−T 1 - T
11 1+(0.302−3.30i)T 1 + (0.302 - 3.30i)T
good3 1+2.27iT−3T2 1 + 2.27iT - 3T^{2}
7 1+3.82T+7T2 1 + 3.82T + 7T^{2}
13 1−4.63iT−13T2 1 - 4.63iT - 13T^{2}
17 1+0.686iT−17T2 1 + 0.686iT - 17T^{2}
19 1−3.82T+19T2 1 - 3.82T + 19T^{2}
23 1+4.18iT−23T2 1 + 4.18iT - 23T^{2}
29 1+3.95iT−29T2 1 + 3.95iT - 29T^{2}
31 1−4.69iT−31T2 1 - 4.69iT - 31T^{2}
37 1+6.34T+37T2 1 + 6.34T + 37T^{2}
41 1−6.74iT−41T2 1 - 6.74iT - 41T^{2}
43 1+2.87T+43T2 1 + 2.87T + 43T^{2}
47 1−12.5iT−47T2 1 - 12.5iT - 47T^{2}
53 1+8.65T+53T2 1 + 8.65T + 53T^{2}
59 1−1.76iT−59T2 1 - 1.76iT - 59T^{2}
61 1+5.32iT−61T2 1 + 5.32iT - 61T^{2}
67 1+4.48iT−67T2 1 + 4.48iT - 67T^{2}
71 1−0.868iT−71T2 1 - 0.868iT - 71T^{2}
73 1+16.1iT−73T2 1 + 16.1iT - 73T^{2}
79 1+16.9T+79T2 1 + 16.9T + 79T^{2}
83 1+4.78T+83T2 1 + 4.78T + 83T^{2}
89 1−5.47T+89T2 1 - 5.47T + 89T^{2}
97 1−17.6T+97T2 1 - 17.6T + 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

−12.43648395423572870459701004852, −11.64103331420624860659926810317, −10.18866428166458405848414956999, −9.357070496291576976293074444836, −7.53495975805134847675627405358, −6.67996324522392562247381617590, −6.24006426723530290361926395734, −4.63994657498015817772683186238, −2.99201337623790660238674931330, −1.74571274180383667961085625767, 3.12752877759680587859872222956, 3.63202919247051812961959745631, 5.29341600032119601072063398567, 5.84202998758659684397507809962, 7.16873502632218701495343664776, 8.699964952455338479617100883210, 9.910224919446315437882780054099, 10.42894991252824074832968314788, 11.49776970444422632388312337601, 12.77305226823515875891606318456

Graph of the ZZ-function along the critical line