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Properties

Label 2-80-4.3-c2-0-0
Degree 22
Conductor 8080
Sign −0.500−0.866i-0.500 - 0.866i
Analytic cond. 2.179842.17984
Root an. cond. 1.476421.47642
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.60i·3-s − 2.23·5-s − 1.32i·7-s − 22.4·9-s + 11.2i·11-s + 17.4·13-s − 12.5i·15-s + 18·17-s + 5.29i·19-s + 7.41·21-s + 15.1i·23-s + 5.00·25-s − 75.1i·27-s + 8.83·29-s − 42.1i·31-s + ⋯
L(s)  = 1  + 1.86i·3-s − 0.447·5-s − 0.189i·7-s − 2.49·9-s + 1.01i·11-s + 1.33·13-s − 0.835i·15-s + 1.05·17-s + 0.278i·19-s + 0.353·21-s + 0.659i·23-s + 0.200·25-s − 2.78i·27-s + 0.304·29-s − 1.36i·31-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((−0.500−0.866i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1) L(s)=((−0.500−0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −0.500−0.866i-0.500 - 0.866i
Analytic conductor: 2.179842.17984
Root analytic conductor: 1.476421.47642
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ80(31,⋅)\chi_{80} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1), −0.500−0.866i)(2,\ 80,\ (\ :1),\ -0.500 - 0.866i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.565442+0.979375i0.565442 + 0.979375i
L(12)L(\frac12) ≈\approx 0.565442+0.979375i0.565442 + 0.979375i
L(2)L(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+2.23T 1 + 2.23T
good3 1−5.60iT−9T2 1 - 5.60iT - 9T^{2}
7 1+1.32iT−49T2 1 + 1.32iT - 49T^{2}
11 1−11.2iT−121T2 1 - 11.2iT - 121T^{2}
13 1−17.4T+169T2 1 - 17.4T + 169T^{2}
17 1−18T+289T2 1 - 18T + 289T^{2}
19 1−5.29iT−361T2 1 - 5.29iT - 361T^{2}
23 1−15.1iT−529T2 1 - 15.1iT - 529T^{2}
29 1−8.83T+841T2 1 - 8.83T + 841T^{2}
31 1+42.1iT−961T2 1 + 42.1iT - 961T^{2}
37 1−33.4T+1.36e3T2 1 - 33.4T + 1.36e3T^{2}
41 1+28.2T+1.68e3T2 1 + 28.2T + 1.68e3T^{2}
43 1+25.3iT−1.84e3T2 1 + 25.3iT - 1.84e3T^{2}
47 1−10.5iT−2.20e3T2 1 - 10.5iT - 2.20e3T^{2}
53 1+28.2T+2.80e3T2 1 + 28.2T + 2.80e3T^{2}
59 1−44.8iT−3.48e3T2 1 - 44.8iT - 3.48e3T^{2}
61 1+77.4T+3.72e3T2 1 + 77.4T + 3.72e3T^{2}
67 1+36.5iT−4.48e3T2 1 + 36.5iT - 4.48e3T^{2}
71 1+97.6iT−5.04e3T2 1 + 97.6iT - 5.04e3T^{2}
73 1−15.6T+5.32e3T2 1 - 15.6T + 5.32e3T^{2}
79 1+112.iT−6.24e3T2 1 + 112. iT - 6.24e3T^{2}
83 1−92.0iT−6.88e3T2 1 - 92.0iT - 6.88e3T^{2}
89 1+59.6T+7.92e3T2 1 + 59.6T + 7.92e3T^{2}
97 1−108.T+9.40e3T2 1 - 108.T + 9.40e3T^{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

−14.89989074908984062418778409037, −13.70757727970655256232665744298, −12.01416148725693276241280105773, −10.98878378402053937591684864785, −10.08604724120706180698544387737, −9.172700467010364855643923581895, −7.889510102332390559813042332016, −5.82781281753287519400421558846, −4.45975206250091000394148278049, −3.47249429946372439329020688835, 1.06904770890671644453700554979, 3.11445754944512377494061572657, 5.76249425566487254658174167997, 6.74053706327825409381364169442, 8.033853818234939696989364129111, 8.680187085025324165906565093447, 10.91787020413572878136658579023, 11.82116438070507778379948410018, 12.72610495171032290374555163714, 13.64826041661203036235922948515

Graph of the ZZ-function along the critical line