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Properties

Label 2-180-20.7-c1-0-12
Degree 22
Conductor 180180
Sign 0.307+0.951i0.307 + 0.951i
Analytic cond. 1.437301.43730
Root an. cond. 1.198871.19887
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  + (1.19 − 0.760i)2-s + (0.844 − 1.81i)4-s + (−0.432 − 2.19i)5-s + (−0.611 + 0.611i)7-s + (−0.371 − 2.80i)8-s + (−2.18 − 2.28i)10-s + 5.12i·11-s + (1.76 − 1.76i)13-s + (−0.264 + 1.19i)14-s + (−2.57 − 3.06i)16-s + (3.76 + 3.76i)17-s + 1.22·19-s + (−4.34 − 1.06i)20-s + (3.89 + 6.11i)22-s + (−1.07 − 1.07i)23-s + ⋯
L(s)  = 1  + (0.843 − 0.537i)2-s + (0.422 − 0.906i)4-s + (−0.193 − 0.981i)5-s + (−0.231 + 0.231i)7-s + (−0.131 − 0.991i)8-s + (−0.690 − 0.723i)10-s + 1.54i·11-s + (0.488 − 0.488i)13-s + (−0.0706 + 0.319i)14-s + (−0.643 − 0.765i)16-s + (0.912 + 0.912i)17-s + 0.280·19-s + (−0.971 − 0.238i)20-s + (0.831 + 1.30i)22-s + (−0.224 − 0.224i)23-s + ⋯

Functional equation

Λ(s)=(180s/2 ΓC(s) L(s)=((0.307+0.951i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(180s/2 ΓC(s+1/2) L(s)=((0.307+0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 180180    =    22⋅32⋅52^{2} \cdot 3^{2} \cdot 5
Sign: 0.307+0.951i0.307 + 0.951i
Analytic conductor: 1.437301.43730
Root analytic conductor: 1.198871.19887
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ180(127,⋅)\chi_{180} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 180, ( :1/2), 0.307+0.951i)(2,\ 180,\ (\ :1/2),\ 0.307 + 0.951i)

Particular Values

L(1)L(1) ≈\approx 1.40002−1.01907i1.40002 - 1.01907i
L(12)L(\frac12) ≈\approx 1.40002−1.01907i1.40002 - 1.01907i
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+(−1.19+0.760i)T 1 + (-1.19 + 0.760i)T
3 1 1
5 1+(0.432+2.19i)T 1 + (0.432 + 2.19i)T
good7 1+(0.611−0.611i)T−7iT2 1 + (0.611 - 0.611i)T - 7iT^{2}
11 1−5.12iT−11T2 1 - 5.12iT - 11T^{2}
13 1+(−1.76+1.76i)T−13iT2 1 + (-1.76 + 1.76i)T - 13iT^{2}
17 1+(−3.76−3.76i)T+17iT2 1 + (-3.76 - 3.76i)T + 17iT^{2}
19 1−1.22T+19T2 1 - 1.22T + 19T^{2}
23 1+(1.07+1.07i)T+23iT2 1 + (1.07 + 1.07i)T + 23iT^{2}
29 1−0.864iT−29T2 1 - 0.864iT - 29T^{2}
31 1−7.81iT−31T2 1 - 7.81iT - 31T^{2}
37 1+(1.76+1.76i)T+37iT2 1 + (1.76 + 1.76i)T + 37iT^{2}
41 1+5.52T+41T2 1 + 5.52T + 41T^{2}
43 1+(6.20+6.20i)T+43iT2 1 + (6.20 + 6.20i)T + 43iT^{2}
47 1+(−2.29+2.29i)T−47iT2 1 + (-2.29 + 2.29i)T - 47iT^{2}
53 1+(−2.62+2.62i)T−53iT2 1 + (-2.62 + 2.62i)T - 53iT^{2}
59 1+0.528T+59T2 1 + 0.528T + 59T^{2}
61 1−4.98T+61T2 1 - 4.98T + 61T^{2}
67 1+(−6.20+6.20i)T−67iT2 1 + (-6.20 + 6.20i)T - 67iT^{2}
71 1+8.10iT−71T2 1 + 8.10iT - 71T^{2}
73 1+(2.25−2.25i)T−73iT2 1 + (2.25 - 2.25i)T - 73iT^{2}
79 1+15.9T+79T2 1 + 15.9T + 79T^{2}
83 1+(7.95+7.95i)T+83iT2 1 + (7.95 + 7.95i)T + 83iT^{2}
89 1−7.25iT−89T2 1 - 7.25iT - 89T^{2}
97 1+(−0.793−0.793i)T+97iT2 1 + (-0.793 - 0.793i)T + 97iT^{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

−12.44765162072276516111970844006, −11.94667747748964375249722095248, −10.48785249998122311784921948008, −9.742311926038661403660322767998, −8.500574689359979173892304161880, −7.10494824111785494056573601109, −5.70191104658883297541916544627, −4.76715708416320154624686582970, −3.54899748688465213528243580191, −1.65964952937428096901794708844, 2.95098694195355075172138316740, 3.84921557091148398694870737657, 5.56623656330314193974881475764, 6.46793810394955261973459951102, 7.47704613072921264889188853072, 8.486326396738225966645734319900, 10.00112740681417922401953134420, 11.38617838431468609852399851364, 11.63028930271591429425122950233, 13.22727555810358714501642760347

Graph of the ZZ-function along the critical line