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Properties

Label 2-220-44.19-c1-0-17
Degree 22
Conductor 220220
Sign 0.122+0.992i0.122 + 0.992i
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  + (−0.317 − 1.37i)2-s + (2.51 − 0.817i)3-s + (−1.79 + 0.874i)4-s + (0.809 + 0.587i)5-s + (−1.92 − 3.21i)6-s + (−0.0373 + 0.114i)7-s + (1.77 + 2.20i)8-s + (3.24 − 2.35i)9-s + (0.553 − 1.30i)10-s + (2.16 − 2.50i)11-s + (−3.81 + 3.67i)12-s + (−3.60 − 4.95i)13-s + (0.170 + 0.0150i)14-s + (2.51 + 0.817i)15-s + (2.47 − 3.14i)16-s + (−3.69 + 5.08i)17-s + ⋯
L(s)  = 1  + (−0.224 − 0.974i)2-s + (1.45 − 0.472i)3-s + (−0.899 + 0.437i)4-s + (0.361 + 0.262i)5-s + (−0.786 − 1.31i)6-s + (−0.0141 + 0.0434i)7-s + (0.627 + 0.778i)8-s + (1.08 − 0.784i)9-s + (0.175 − 0.411i)10-s + (0.654 − 0.756i)11-s + (−1.10 + 1.05i)12-s + (−0.998 − 1.37i)13-s + (0.0454 + 0.00401i)14-s + (0.649 + 0.211i)15-s + (0.617 − 0.786i)16-s + (−0.895 + 1.23i)17-s + ⋯

Functional equation

Λ(s)=(220s/2 ΓC(s) L(s)=((0.122+0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((0.122+0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: 0.122+0.992i0.122 + 0.992i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(151,⋅)\chi_{220} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 220, ( :1/2), 0.122+0.992i)(2,\ 220,\ (\ :1/2),\ 0.122 + 0.992i)

Particular Values

L(1)L(1) ≈\approx 1.19675−1.05796i1.19675 - 1.05796i
L(12)L(\frac12) ≈\approx 1.19675−1.05796i1.19675 - 1.05796i
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+(0.317+1.37i)T 1 + (0.317 + 1.37i)T
5 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1+(−2.16+2.50i)T 1 + (-2.16 + 2.50i)T
good3 1+(−2.51+0.817i)T+(2.42−1.76i)T2 1 + (-2.51 + 0.817i)T + (2.42 - 1.76i)T^{2}
7 1+(0.0373−0.114i)T+(−5.66−4.11i)T2 1 + (0.0373 - 0.114i)T + (-5.66 - 4.11i)T^{2}
13 1+(3.60+4.95i)T+(−4.01+12.3i)T2 1 + (3.60 + 4.95i)T + (-4.01 + 12.3i)T^{2}
17 1+(3.69−5.08i)T+(−5.25−16.1i)T2 1 + (3.69 - 5.08i)T + (-5.25 - 16.1i)T^{2}
19 1+(−2.12−6.54i)T+(−15.3+11.1i)T2 1 + (-2.12 - 6.54i)T + (-15.3 + 11.1i)T^{2}
23 1−0.892iT−23T2 1 - 0.892iT - 23T^{2}
29 1+(8.01+2.60i)T+(23.4+17.0i)T2 1 + (8.01 + 2.60i)T + (23.4 + 17.0i)T^{2}
31 1+(−2.18−3.00i)T+(−9.57+29.4i)T2 1 + (-2.18 - 3.00i)T + (-9.57 + 29.4i)T^{2}
37 1+(−0.0956+0.294i)T+(−29.9−21.7i)T2 1 + (-0.0956 + 0.294i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.65+0.538i)T+(33.1−24.0i)T2 1 + (-1.65 + 0.538i)T + (33.1 - 24.0i)T^{2}
43 1−2.73T+43T2 1 - 2.73T + 43T^{2}
47 1+(0.176−0.0572i)T+(38.0−27.6i)T2 1 + (0.176 - 0.0572i)T + (38.0 - 27.6i)T^{2}
53 1+(5.28−3.83i)T+(16.3−50.4i)T2 1 + (5.28 - 3.83i)T + (16.3 - 50.4i)T^{2}
59 1+(−1.79−0.583i)T+(47.7+34.6i)T2 1 + (-1.79 - 0.583i)T + (47.7 + 34.6i)T^{2}
61 1+(−0.785+1.08i)T+(−18.8−58.0i)T2 1 + (-0.785 + 1.08i)T + (-18.8 - 58.0i)T^{2}
67 1+10.2iT−67T2 1 + 10.2iT - 67T^{2}
71 1+(1.37−1.89i)T+(−21.9−67.5i)T2 1 + (1.37 - 1.89i)T + (-21.9 - 67.5i)T^{2}
73 1+(−0.938−0.305i)T+(59.0+42.9i)T2 1 + (-0.938 - 0.305i)T + (59.0 + 42.9i)T^{2}
79 1+(−3.76+2.73i)T+(24.4−75.1i)T2 1 + (-3.76 + 2.73i)T + (24.4 - 75.1i)T^{2}
83 1+(4.72+3.43i)T+(25.6+78.9i)T2 1 + (4.72 + 3.43i)T + (25.6 + 78.9i)T^{2}
89 1+12.6T+89T2 1 + 12.6T + 89T^{2}
97 1+(−4.90+3.56i)T+(29.9−92.2i)T2 1 + (-4.90 + 3.56i)T + (29.9 - 92.2i)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

−12.32599199729321091847118568278, −10.96123129576630163621910689921, −10.00992204999340640410827275038, −9.159054515339040282234077762175, −8.246442008997446956553635520674, −7.55515754611158899479011044035, −5.78642745733490598131442511975, −3.86920646866479806392111468243, −2.94736250963279701388658599669, −1.74163957948841734115940594688, 2.29673661208280075467558209571, 4.17464197974398390641968060244, 4.93979583381904405025878302397, 6.84231427428518995568195784674, 7.41350583842732264545700507194, 8.890159734195619177003032644500, 9.283975136801755883764257174992, 9.803658319139354316446761733422, 11.51439553018973332085834794095, 13.02249218893413163287858984033

Graph of the ZZ-function along the critical line