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Properties

Label 2-2160-4.3-c2-0-54
Degree 22
Conductor 21602160
Sign ii
Analytic cond. 58.855758.8557
Root an. cond. 7.671747.67174
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  + 2.23·5-s − 7.74i·7-s + 6.92i·11-s + 22·13-s − 6.70·17-s − 27.1i·19-s − 29.4i·23-s + 5.00·25-s − 40.2·29-s + 19.3i·31-s − 17.3i·35-s + 2·37-s + 53.6·41-s − 15.4i·43-s + 13.8i·47-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.10i·7-s + 0.629i·11-s + 1.69·13-s − 0.394·17-s − 1.42i·19-s − 1.28i·23-s + 0.200·25-s − 1.38·29-s + 0.624i·31-s − 0.494i·35-s + 0.0540·37-s + 1.30·41-s − 0.360i·43-s + 0.294i·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=(i Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: ii
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(271,⋅)\chi_{2160} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), i)(2,\ 2160,\ (\ :1),\ i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.0979505362.097950536
L(12)L(\frac12) ≈\approx 2.0979505362.097950536
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
3 1 1
5 1−2.23T 1 - 2.23T
good7 1+7.74iT−49T2 1 + 7.74iT - 49T^{2}
11 1−6.92iT−121T2 1 - 6.92iT - 121T^{2}
13 1−22T+169T2 1 - 22T + 169T^{2}
17 1+6.70T+289T2 1 + 6.70T + 289T^{2}
19 1+27.1iT−361T2 1 + 27.1iT - 361T^{2}
23 1+29.4iT−529T2 1 + 29.4iT - 529T^{2}
29 1+40.2T+841T2 1 + 40.2T + 841T^{2}
31 1−19.3iT−961T2 1 - 19.3iT - 961T^{2}
37 1−2T+1.36e3T2 1 - 2T + 1.36e3T^{2}
41 1−53.6T+1.68e3T2 1 - 53.6T + 1.68e3T^{2}
43 1+15.4iT−1.84e3T2 1 + 15.4iT - 1.84e3T^{2}
47 1−13.8iT−2.20e3T2 1 - 13.8iT - 2.20e3T^{2}
53 1+6.70T+2.80e3T2 1 + 6.70T + 2.80e3T^{2}
59 1+24.2iT−3.48e3T2 1 + 24.2iT - 3.48e3T^{2}
61 1+31T+3.72e3T2 1 + 31T + 3.72e3T^{2}
67 1+23.2iT−4.48e3T2 1 + 23.2iT - 4.48e3T^{2}
71 1−110.iT−5.04e3T2 1 - 110. iT - 5.04e3T^{2}
73 1−76T+5.32e3T2 1 - 76T + 5.32e3T^{2}
79 1+19.3iT−6.24e3T2 1 + 19.3iT - 6.24e3T^{2}
83 1+129.iT−6.88e3T2 1 + 129. iT - 6.88e3T^{2}
89 1+53.6T+7.92e3T2 1 + 53.6T + 7.92e3T^{2}
97 1+32T+9.40e3T2 1 + 32T + 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

−8.830173435745243727517795151459, −7.903207198052495371726304631950, −6.99047278180165918126537386472, −6.53081256663946218857263983431, −5.55999800126947305266471731331, −4.50289393042389497749177703367, −3.92465945451352093478958912425, −2.76823742969333761932864280262, −1.59680169695039543470418274483, −0.54245034668820873482215305507, 1.25561480993411243329743980707, 2.16377751141339772657054805243, 3.34257168041660878304482728432, 4.03423783065252859800427068087, 5.59331053482706433376081585094, 5.72279026492969566119567566357, 6.48531317644886442151920960884, 7.75258800922781567141771680335, 8.344773695375295859534147041636, 9.175919213427991534650467254371

Graph of the ZZ-function along the critical line