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Properties

Label 2-180-60.47-c2-0-9
Degree 22
Conductor 180180
Sign 0.338−0.940i0.338 - 0.940i
Analytic cond. 4.904644.90464
Root an. cond. 2.214642.21464
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  + (1.97 + 0.321i)2-s + (3.79 + 1.26i)4-s + (0.196 + 4.99i)5-s + (−8.42 + 8.42i)7-s + (7.08 + 3.72i)8-s + (−1.21 + 9.92i)10-s + 0.926·11-s + (8.10 − 8.10i)13-s + (−19.3 + 13.9i)14-s + (12.7 + 9.62i)16-s + (20.7 − 20.7i)17-s − 13.8·19-s + (−5.59 + 19.2i)20-s + (1.82 + 0.297i)22-s + (22.9 − 22.9i)23-s + ⋯
L(s)  = 1  + (0.987 + 0.160i)2-s + (0.948 + 0.317i)4-s + (0.0393 + 0.999i)5-s + (−1.20 + 1.20i)7-s + (0.885 + 0.465i)8-s + (−0.121 + 0.992i)10-s + 0.0841·11-s + (0.623 − 0.623i)13-s + (−1.38 + 0.994i)14-s + (0.798 + 0.601i)16-s + (1.21 − 1.21i)17-s − 0.728·19-s + (−0.279 + 0.960i)20-s + (0.0830 + 0.0135i)22-s + (0.999 − 0.999i)23-s + ⋯

Functional equation

Λ(s)=(180s/2 ΓC(s) L(s)=((0.338−0.940i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(180s/2 ΓC(s+1) L(s)=((0.338−0.940i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 180180    =    22⋅32⋅52^{2} \cdot 3^{2} \cdot 5
Sign: 0.338−0.940i0.338 - 0.940i
Analytic conductor: 4.904644.90464
Root analytic conductor: 2.214642.21464
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ180(107,⋅)\chi_{180} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 180, ( :1), 0.338−0.940i)(2,\ 180,\ (\ :1),\ 0.338 - 0.940i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.05557+1.44506i2.05557 + 1.44506i
L(12)L(\frac12) ≈\approx 2.05557+1.44506i2.05557 + 1.44506i
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+(−1.97−0.321i)T 1 + (-1.97 - 0.321i)T
3 1 1
5 1+(−0.196−4.99i)T 1 + (-0.196 - 4.99i)T
good7 1+(8.42−8.42i)T−49iT2 1 + (8.42 - 8.42i)T - 49iT^{2}
11 1−0.926T+121T2 1 - 0.926T + 121T^{2}
13 1+(−8.10+8.10i)T−169iT2 1 + (-8.10 + 8.10i)T - 169iT^{2}
17 1+(−20.7+20.7i)T−289iT2 1 + (-20.7 + 20.7i)T - 289iT^{2}
19 1+13.8T+361T2 1 + 13.8T + 361T^{2}
23 1+(−22.9+22.9i)T−529iT2 1 + (-22.9 + 22.9i)T - 529iT^{2}
29 1−12.0T+841T2 1 - 12.0T + 841T^{2}
31 1−27.6iT−961T2 1 - 27.6iT - 961T^{2}
37 1+(−14.0−14.0i)T+1.36e3iT2 1 + (-14.0 - 14.0i)T + 1.36e3iT^{2}
41 1−10.4iT−1.68e3T2 1 - 10.4iT - 1.68e3T^{2}
43 1+(44.6+44.6i)T+1.84e3iT2 1 + (44.6 + 44.6i)T + 1.84e3iT^{2}
47 1+(8.50+8.50i)T+2.20e3iT2 1 + (8.50 + 8.50i)T + 2.20e3iT^{2}
53 1+(7.70+7.70i)T+2.80e3iT2 1 + (7.70 + 7.70i)T + 2.80e3iT^{2}
59 1−12.6iT−3.48e3T2 1 - 12.6iT - 3.48e3T^{2}
61 1+13.7T+3.72e3T2 1 + 13.7T + 3.72e3T^{2}
67 1+(−75.3+75.3i)T−4.48e3iT2 1 + (-75.3 + 75.3i)T - 4.48e3iT^{2}
71 1+13.5T+5.04e3T2 1 + 13.5T + 5.04e3T^{2}
73 1+(43.6−43.6i)T−5.32e3iT2 1 + (43.6 - 43.6i)T - 5.32e3iT^{2}
79 1−71.8T+6.24e3T2 1 - 71.8T + 6.24e3T^{2}
83 1+(21.6−21.6i)T−6.88e3iT2 1 + (21.6 - 21.6i)T - 6.88e3iT^{2}
89 1+98.6T+7.92e3T2 1 + 98.6T + 7.92e3T^{2}
97 1+(27.6+27.6i)T+9.40e3iT2 1 + (27.6 + 27.6i)T + 9.40e3iT^{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

−12.62150993695955965134014942311, −11.91512149591311785392135354701, −10.80791584523132748900010537822, −9.837477757679865334745240322437, −8.423751241061476255953665259901, −6.97610598273763999279740339846, −6.26492924317235747883358040843, −5.25592948810283913383735222094, −3.35219707055608948171136186004, −2.69892354521630943315189029738, 1.27230976356102384405372460237, 3.49682128438499588379062787015, 4.30201166560570249179276985096, 5.77209815290378162704273707937, 6.70955832974120708705070813793, 7.945263257241490827093803455140, 9.473721488731506901369571263489, 10.34605713940288384891387334830, 11.42707084797511519454788416479, 12.63043419369383067369930684028

Graph of the ZZ-function along the critical line