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Properties

Label 2-220-55.2-c1-0-0
Degree 22
Conductor 220220
Sign −0.632−0.774i-0.632 - 0.774i
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  + (−0.467 + 2.95i)3-s + (−0.111 + 2.23i)5-s + (2.35 − 0.372i)7-s + (−5.63 − 1.83i)9-s + (0.368 − 3.29i)11-s + (−4.91 + 2.50i)13-s + (−6.53 − 1.37i)15-s + (5.13 + 2.61i)17-s + (−2.00 + 1.45i)19-s + 7.11i·21-s + (4.85 − 4.85i)23-s + (−4.97 − 0.499i)25-s + (3.96 − 7.78i)27-s + (4.14 + 3.01i)29-s + (0.457 − 1.40i)31-s + ⋯
L(s)  = 1  + (−0.269 + 1.70i)3-s + (−0.0500 + 0.998i)5-s + (0.889 − 0.140i)7-s + (−1.87 − 0.610i)9-s + (0.111 − 0.993i)11-s + (−1.36 + 0.694i)13-s + (−1.68 − 0.354i)15-s + (1.24 + 0.634i)17-s + (−0.460 + 0.334i)19-s + 1.55i·21-s + (1.01 − 1.01i)23-s + (−0.994 − 0.0999i)25-s + (0.762 − 1.49i)27-s + (0.770 + 0.559i)29-s + (0.0821 − 0.252i)31-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.632−0.774i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((−0.632−0.774i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.632−0.774i-0.632 - 0.774i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(57,⋅)\chi_{220} (57, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), −0.632−0.774i)(2,\ 220,\ (\ :1/2),\ -0.632 - 0.774i)

Particular Values

L(1)L(1) ≈\approx 0.472664+0.995593i0.472664 + 0.995593i
L(12)L(\frac12) ≈\approx 0.472664+0.995593i0.472664 + 0.995593i
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
5 1+(0.111−2.23i)T 1 + (0.111 - 2.23i)T
11 1+(−0.368+3.29i)T 1 + (-0.368 + 3.29i)T
good3 1+(0.467−2.95i)T+(−2.85−0.927i)T2 1 + (0.467 - 2.95i)T + (-2.85 - 0.927i)T^{2}
7 1+(−2.35+0.372i)T+(6.65−2.16i)T2 1 + (-2.35 + 0.372i)T + (6.65 - 2.16i)T^{2}
13 1+(4.91−2.50i)T+(7.64−10.5i)T2 1 + (4.91 - 2.50i)T + (7.64 - 10.5i)T^{2}
17 1+(−5.13−2.61i)T+(9.99+13.7i)T2 1 + (-5.13 - 2.61i)T + (9.99 + 13.7i)T^{2}
19 1+(2.00−1.45i)T+(5.87−18.0i)T2 1 + (2.00 - 1.45i)T + (5.87 - 18.0i)T^{2}
23 1+(−4.85+4.85i)T−23iT2 1 + (-4.85 + 4.85i)T - 23iT^{2}
29 1+(−4.14−3.01i)T+(8.96+27.5i)T2 1 + (-4.14 - 3.01i)T + (8.96 + 27.5i)T^{2}
31 1+(−0.457+1.40i)T+(−25.0−18.2i)T2 1 + (-0.457 + 1.40i)T + (-25.0 - 18.2i)T^{2}
37 1+(0.103+0.654i)T+(−35.1+11.4i)T2 1 + (0.103 + 0.654i)T + (-35.1 + 11.4i)T^{2}
41 1+(−0.176−0.242i)T+(−12.6+38.9i)T2 1 + (-0.176 - 0.242i)T + (-12.6 + 38.9i)T^{2}
43 1+(−6.73−6.73i)T+43iT2 1 + (-6.73 - 6.73i)T + 43iT^{2}
47 1+(−7.46−1.18i)T+(44.6+14.5i)T2 1 + (-7.46 - 1.18i)T + (44.6 + 14.5i)T^{2}
53 1+(0.877+1.72i)T+(−31.1+42.8i)T2 1 + (0.877 + 1.72i)T + (-31.1 + 42.8i)T^{2}
59 1+(−0.419+0.576i)T+(−18.2−56.1i)T2 1 + (-0.419 + 0.576i)T + (-18.2 - 56.1i)T^{2}
61 1+(−5.32+1.73i)T+(49.3−35.8i)T2 1 + (-5.32 + 1.73i)T + (49.3 - 35.8i)T^{2}
67 1+(6.03+6.03i)T+67iT2 1 + (6.03 + 6.03i)T + 67iT^{2}
71 1+(1.48+4.56i)T+(−57.4+41.7i)T2 1 + (1.48 + 4.56i)T + (-57.4 + 41.7i)T^{2}
73 1+(−0.311−1.96i)T+(−69.4+22.5i)T2 1 + (-0.311 - 1.96i)T + (-69.4 + 22.5i)T^{2}
79 1+(−2.20+6.79i)T+(−63.9−46.4i)T2 1 + (-2.20 + 6.79i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.000475−0.000934i)T+(−48.7−67.1i)T2 1 + (0.000475 - 0.000934i)T + (-48.7 - 67.1i)T^{2}
89 1−4.41iT−89T2 1 - 4.41iT - 89T^{2}
97 1+(7.94−4.05i)T+(57.0−78.4i)T2 1 + (7.94 - 4.05i)T + (57.0 - 78.4i)T^{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

−12.25975165966916893169898226796, −11.27004657125575571835257821617, −10.68494472462660724429169994890, −9.993195851111424057436043227681, −8.921390793847294828500506837242, −7.76431224996133310476189314991, −6.28274525462595191127994662185, −5.10840908601421569395961569610, −4.11992434092538025913907161252, −2.91066225364259885643074043508, 1.04498458614905352770579623825, 2.37203406668290061412555987852, 4.85674927802972792129147083606, 5.60163590168205481018852504757, 7.26316351367758729295661453062, 7.61954850139093181639833430232, 8.686462829713609385039696736573, 9.929307734426621441948663210516, 11.48790785070983068499519380221, 12.23415660596411427188098462316

Graph of the ZZ-function along the critical line