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Properties

Label 2-2160-15.14-c2-0-66
Degree 22
Conductor 21602160
Sign 0.599+0.799i0.599 + 0.799i
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  + (4 − 3i)5-s − 6i·7-s + 21i·11-s − 15i·13-s + 23·17-s − 14·19-s + 7·23-s + (7 − 24i)25-s − 3i·29-s + 25·31-s + (−18 − 24i)35-s + 54i·37-s − 24i·41-s − 15i·43-s + 49·47-s + ⋯
L(s)  = 1  + (0.800 − 0.600i)5-s − 0.857i·7-s + 1.90i·11-s − 1.15i·13-s + 1.35·17-s − 0.736·19-s + 0.304·23-s + (0.280 − 0.959i)25-s − 0.103i·29-s + 0.806·31-s + (−0.514 − 0.685i)35-s + 1.45i·37-s − 0.585i·41-s − 0.348i·43-s + 1.04·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.599+0.799i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((0.599+0.799i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.599+0.799i0.599 + 0.799i
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), 0.599+0.799i)(2,\ 2160,\ (\ :1),\ 0.599 + 0.799i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.5312098062.531209806
L(12)L(\frac12) ≈\approx 2.5312098062.531209806
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+(−4+3i)T 1 + (-4 + 3i)T
good7 1+6iT−49T2 1 + 6iT - 49T^{2}
11 1−21iT−121T2 1 - 21iT - 121T^{2}
13 1+15iT−169T2 1 + 15iT - 169T^{2}
17 1−23T+289T2 1 - 23T + 289T^{2}
19 1+14T+361T2 1 + 14T + 361T^{2}
23 1−7T+529T2 1 - 7T + 529T^{2}
29 1+3iT−841T2 1 + 3iT - 841T^{2}
31 1−25T+961T2 1 - 25T + 961T^{2}
37 1−54iT−1.36e3T2 1 - 54iT - 1.36e3T^{2}
41 1+24iT−1.68e3T2 1 + 24iT - 1.68e3T^{2}
43 1+15iT−1.84e3T2 1 + 15iT - 1.84e3T^{2}
47 1−49T+2.20e3T2 1 - 49T + 2.20e3T^{2}
53 1−14T+2.80e3T2 1 - 14T + 2.80e3T^{2}
59 1+30iT−3.48e3T2 1 + 30iT - 3.48e3T^{2}
61 1−44T+3.72e3T2 1 - 44T + 3.72e3T^{2}
67 1+66iT−4.48e3T2 1 + 66iT - 4.48e3T^{2}
71 1−18iT−5.04e3T2 1 - 18iT - 5.04e3T^{2}
73 1−5.32e3T2 1 - 5.32e3T^{2}
79 1−37T+6.24e3T2 1 - 37T + 6.24e3T^{2}
83 1+116T+6.88e3T2 1 + 116T + 6.88e3T^{2}
89 1+126iT−7.92e3T2 1 + 126iT - 7.92e3T^{2}
97 1−78iT−9.40e3T2 1 - 78iT - 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

−8.788992994663698616355236674522, −7.930803258645152067740069378333, −7.29470958958619771238313617964, −6.46575682133373419199520583446, −5.47543344023529242908169832353, −4.81434126231532467440711517544, −4.02181385679837111083763660890, −2.78015530362325746881189242271, −1.70433687238204907052358108119, −0.73857156632404016968590844807, 1.03800421838750038478354674208, 2.28922265167572238473264429065, 3.03131882103692379178825404630, 3.97941813767163814965820891214, 5.35126404017434479731769543747, 5.88107980852728526465272388992, 6.44355303341868154947900732554, 7.40643734124517462809278039862, 8.481322111909778583747951604464, 8.928788149191546541089377415239

Graph of the ZZ-function along the critical line