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Properties

Label 2-2160-15.14-c2-0-10
Degree 22
Conductor 21602160
Sign −0.905+0.424i-0.905 + 0.424i
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  + (2.12 + 4.52i)5-s + 4.02i·7-s + 11.5i·11-s + 13.5i·13-s − 4.76·17-s − 8.28·19-s − 33.4·23-s + (−15.9 + 19.2i)25-s − 21.3i·29-s − 51.2·31-s + (−18.2 + 8.54i)35-s − 1.30i·37-s + 0.277i·41-s − 5.16i·43-s + 49.9·47-s + ⋯
L(s)  = 1  + (0.424 + 0.905i)5-s + 0.574i·7-s + 1.05i·11-s + 1.04i·13-s − 0.280·17-s − 0.435·19-s − 1.45·23-s + (−0.638 + 0.769i)25-s − 0.735i·29-s − 1.65·31-s + (−0.520 + 0.244i)35-s − 0.0351i·37-s + 0.00676i·41-s − 0.120i·43-s + 1.06·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.905+0.424i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.905+0.424i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.905+0.424i-0.905 + 0.424i
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), −0.905+0.424i)(2,\ 2160,\ (\ :1),\ -0.905 + 0.424i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.81215720280.8121572028
L(12)L(\frac12) ≈\approx 0.81215720280.8121572028
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+(−2.12−4.52i)T 1 + (-2.12 - 4.52i)T
good7 1−4.02iT−49T2 1 - 4.02iT - 49T^{2}
11 1−11.5iT−121T2 1 - 11.5iT - 121T^{2}
13 1−13.5iT−169T2 1 - 13.5iT - 169T^{2}
17 1+4.76T+289T2 1 + 4.76T + 289T^{2}
19 1+8.28T+361T2 1 + 8.28T + 361T^{2}
23 1+33.4T+529T2 1 + 33.4T + 529T^{2}
29 1+21.3iT−841T2 1 + 21.3iT - 841T^{2}
31 1+51.2T+961T2 1 + 51.2T + 961T^{2}
37 1+1.30iT−1.36e3T2 1 + 1.30iT - 1.36e3T^{2}
41 1−0.277iT−1.68e3T2 1 - 0.277iT - 1.68e3T^{2}
43 1+5.16iT−1.84e3T2 1 + 5.16iT - 1.84e3T^{2}
47 1−49.9T+2.20e3T2 1 - 49.9T + 2.20e3T^{2}
53 1+14.5T+2.80e3T2 1 + 14.5T + 2.80e3T^{2}
59 1+56.4iT−3.48e3T2 1 + 56.4iT - 3.48e3T^{2}
61 1−108.T+3.72e3T2 1 - 108.T + 3.72e3T^{2}
67 1+61.7iT−4.48e3T2 1 + 61.7iT - 4.48e3T^{2}
71 1+63.6iT−5.04e3T2 1 + 63.6iT - 5.04e3T^{2}
73 1+3.03iT−5.32e3T2 1 + 3.03iT - 5.32e3T^{2}
79 1+24.1T+6.24e3T2 1 + 24.1T + 6.24e3T^{2}
83 1+54.1T+6.88e3T2 1 + 54.1T + 6.88e3T^{2}
89 1−19.2iT−7.92e3T2 1 - 19.2iT - 7.92e3T^{2}
97 1−179.iT−9.40e3T2 1 - 179. iT - 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

−9.468942139186550315596957911535, −8.664066791772652052125381359855, −7.62793865620666754135447029277, −6.97615306266722119683161832796, −6.24933430482178629646097186096, −5.53153331500417462745243457582, −4.41017376121900751276880097266, −3.64451358214034109281132432799, −2.20766979950709360168945402489, −2.00528621537479012284813739118, 0.19687226707951845148685151685, 1.17360126126974376797046651840, 2.36959833930904987349083841007, 3.60668067750953431154251723023, 4.30898883920631730744498620731, 5.54259075627942610497419023492, 5.75037831904661811218482381715, 6.91619137693623628867058378812, 7.83124468597332589317321755004, 8.509219732143335628058264221750

Graph of the ZZ-function along the critical line